b
KATHOLIEKE UNIVERSITEIT LEUVEN
Faculteit Wetenschappen
Departement Wiskunde
Afdeling Meetkunde
Ruled Surfaces of Weingarten Type
in Minkowski 3-space
Wijarn Sodsiri
Proefschrift ingediend tot het
Promotor:
Prof. dr. Franki Dillen
behalen van de graad van
Doctor in de Wetenschappen
2005
b
KATHOLIEKE UNIVERSITEIT LEUVEN
Faculteit Wetenschappen
Departement Wiskunde
Afdeling Meetkunde
Ruled Surfaces of Weingarten Type
in Minkowski 3-space
Proefschrift ingediend tot het
Jury:
behalen van de graad van
Prof. E. Boeckx (K.U. Leuven)
Prof. F. Dillen (Promotor, K.U. Leuven)
Prof. I. Van de Woestijne (K.U. Brussel)
Prof. L. Verstraelen (K.U. Leuven)
Prof. L. Vrancken (U. Valenciennes, France)
2005
Doctor in de Wetenschappen
door
Wijarn SODSIRI
Acknowledgements
I was first introduced to the Katholieke Universiteit Leuven (K.U. Leuven) by Professor Paul Tobback, a former director of the international
relations office of K.U. Leuven, during his visit to Khon Kaen University,
Thailand, in 1998. At that time Thailand needed differential geometers
for academic activities and I was looking for a school to do my PhD in
topology or geometry. After getting some information from the websites
of K.U. Leuven and of the Geometry research group I decided to attend
a PhD program in this research group.
It was Professor Paul Dhooghe, a former head of the research group,
who kindly accepted to be my thesis supervisor (or promotor in Dutch)
during my first two years here letting me find a way of research. In the
end, “ruled Weingarten surfaces in Minkowski 3-space” is the topic for
my PhD research which was taken into my account by Professor Franki
Dillen, my present promotor.
Studying differential geometry from scratch here, I have found that
some geometric notions are very difficult for me to comprehend, but they
have never made me give up. Fortunately, many people have been glad to
answer my innocent questions and provide me the methods of geometric
thinking which are not available at any supermarket. One of the hands I
have received is Professor Eric Boeckx’s excellent English translation of
i
ii
Acknowledgements
G. Stamou’s article [31], entitled Regelflächen vom Weingarten-typ, which
at last becomes an oasis for this dissertation.
I am greatly indebted to Professor Franki Dillen, my promotor, for his
thoughtful and helpful advice, which he offered tirelessly throughout the
preparation and writing of this dissertation. I would also like to thank
everyone who kindly gave help to me, especially Professor Eric Boeckx,
Professor Wolfgang Kühnel (U. Stuttgart), Professor Friedrich Manhart
(T.U. Wien), Professor Lieven Vanhecke, Professor Leopold Verstraelen,
Dr. Stefan Haesen, Dr. Ying Lu, and Dr. Gerd Verbouwe as well as the
members of our research group, both former and present. My thanks must
also go to Ms. Wendy Goemans, who not only read the early drafts of
this dissertation with incredible thoroughness and found many misprints,
but also checked many of the computations therein.
This dissertation would not exist without generous suggestions and
comments of the dissertation committee members: Professor Eric Boeckx
(K.U. Leuven), Professor Franki Dillen (K.U. Leuven), Professor Ignace
Van de Woestijne (K.U. Brussel), Professor Leopold Verstraelen (K.U.
Leuven), and Professor Luc Vrancken (U. Valenciennes, France). I am
very grateful to all of them.
Last but not least, I wish to sincerely express my gratitude to Khon
Kaen University and the Thai government for the grants I have received
during my PhD study.
Leuven (Heverlee)
March 2005
Wijarn Sodsiri
Contents
Acknowledgements
i
List of Figures
vii
Introduction
ix
1 Preliminaries
1.1
1.2
1
Vector Algebra in E31 . . . . . . . . . . . . . . . . . . . . .
Frenet Formulas in
R3
E31
2
. . . . . . . . . . . . . . . . . . . .
3
1.3
Surfaces in
. . . . . . . . . . . . . . . . . . . . . . . . .
4
1.4
Ruled Surfaces . . . . . . . . . . . . . . . . . . . . . . . .
6
1.5
Shape Operators . . . . . . . . . . . . . . . . . . . . . . .
8
1.6
Gauss and Weingarten Equations in E31
. . . . . . . . . .
11
1.7
Some Differential Operators . . . . . . . . . . . . . . . . .
15
1.8
Divergence Theorem . . . . . . . . . . . . . . . . . . . . .
17
2 Ricci Calculus
2.1
19
Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.1.1
Basic Algebra . . . . . . . . . . . . . . . . . . . . .
19
2.1.2
Tensor Fields . . . . . . . . . . . . . . . . . . . . .
21
2.1.3
Interpretations . . . . . . . . . . . . . . . . . . . .
22
iii
iv
Contents
2.1.4
Tensor Components . . . . . . . . . . . . . . . . .
23
2.2
Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . .
23
2.3
Raising and Lowering Indices . . . . . . . . . . . . . . . .
26
2.4
Fiber Metrics . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.5
Parallel Tensor Fields . . . . . . . . . . . . . . . . . . . .
30
2.6
The Ricci Lemma . . . . . . . . . . . . . . . . . . . . . . .
31
2.7
Contraction . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3 Matrix Groups
35
3.1
Groups of Matrices . . . . . . . . . . . . . . . . . . . . . .
35
3.2
Groups of Matrices as Metric Spaces . . . . . . . . . . . .
36
3.3
Matrix Groups . . . . . . . . . . . . . . . . . . . . . . . .
39
3.4
Some Important Examples . . . . . . . . . . . . . . . . . .
39
3.5
Lorentzian Groups . . . . . . . . . . . . . . . . . . . . . .
42
3.6
One-parameter Groups in Matrix Groups . . . . . . . . .
47
4 Lorentzian Motions in E31
49
4.1
Groups of Linear Lorentzian Rotations . . . . . . . . . . .
49
4.2
Groups of Lorentzian Motions in E31 . . . . . . . . . . . .
54
5 Curvatures
61
5.1
The Gaussian and Mean Curvatures . . . . . . . . . . . .
61
5.2
The Second Gaussian and Second Mean Curvatures . . . .
67
5.3
Curvatures of Ruled Surfaces in E31 . . . . . . . . . . . . .
79
6 Ruled Weingarten Surfaces
85
6.1
Helicoidal Ruled Surfaces . . . . . . . . . . . . . . . . . .
85
6.2
Ruled Weingarten Surfaces in E3 . . . . . . . . . . . . . .
90
6.3
Ruled Weingarten Surfaces in E31 . . . . . . . . . . . . . .
91
6.4
Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Contents
v
7 Ruled Linear Weingarten Surfaces
109
E3
7.1
Ruled Linear Weingarten Surfaces in
. . . . . . . . . . 109
7.2
Ruled Linear Weingarten Surfaces in E31 . . . . . . . . . . 111
A Some Results from Real Analysis
117
B Proof of Lemma 5.14
123
C Ruled Surfaces in E31
131
D Julius Weingarten
141
Bibliography
145
Index of Symbols
149
Index
151
List of Figures
C.1 Helicoidal ruled surface x(s, t) = (s+t, cos s−t sin s, sin s+
t cos s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
C.2 Helicoid of the 1st kind x(s, t) = (as, t cos s, t sin s) . . . . 132
C.3 Helicoid of the 2nd kind x = y tanh(z/a) . . . . . . . . . . 133
C.4 Helicoid of the 3rd kind y = x tanh(z/a) . . . . . . . . . . 134
C.5 Conjugate of Enneper’s surface of the 2nd kind or Cayley–
Lie minimal ruled surface of third degree 6a2 (x + y) =
¢
¡
(x − y) 6az − (x − y)2 . . . . . . . . . . . . . . . . . . . 135
x2 − y 2
C.6 Nomizu–Sasaki surface z =
+ b ln |x − y| . . . . . 136
2a
C.7 Conoid of the 1st kind x(s, t) = (exp(s), t cos s, t sin s) with
the curve c(s) = (exp(s), cos s, sin s) . . . . . . . . . . . . 137
C.8 Conoid of the 2nd kind x(s, t) = (t sinh s, t cosh s, s3 + s)
with the curve c(s) = (2 sinh s, 2 cosh s, s3 + s) . . . . . . . 138
C.9 Conoid of the 3rd kind x(s, t) = (t cosh s, t sinh s, s3 + s)
with the curve c(s) = (3 cosh s, 3 sinh s, s3 + s) . . . . . . . 139
D.1 Julius Weingarten . . . . . . . . . . . . . . . . . . . . . . 141
vii
Introduction
Throughout this dissertation, the surfaces in question are smooth and
connected. We will abuse the term “nondevelopable,” and by a nondevelopable surface we mean a surface free of points of vanishing Gaussian
curvature.
Classically, a Weingarten surface (or briefly, a W-surface) is a surface
on which there is a nontrivial functional relation Φ(k1 , k2 ) = 0 between
its principal curvatures k1 and k2 , or, equivalently, there is a nontrivial functional relation Φ(K, H) = 0 between its Gaussian curvature K
and mean curvature H. The existence of a nontrivial functional relation
Φ(A, B) = 0 such that Φ is of class C1 is equivalent to the vanishing of
the corresponding Jacobian determinant, namely,
∂(A, B)
≡ 0,
∂(s, t)
where (A, B) = (k1 , k2 ) or (K, H). Examples of Weingarten surfaces include surfaces of constant mean curvature (e.g., minimal surfaces, which
are still the targets of much research), surfaces of constant Gaussian curvature (e.g., spheres and developable surfaces), surfaces of revolution, etc.
Developable ruled surfaces are those which are intrinsically flat, namely, they are of vanishing Gaussian curvature. These surfaces are characterized by the constancy of the tangent plane along each ruling. A classical
ix
x
Introduction
result in differential geometry states that an open and dense subset of a
developable ruled surface consists of pieces of planes, cylinders, cones and
tangent surfaces. This is exactly the same statement for both Euclidean
and Minkowski spaces as ambient space, but we have to assume that the
tangent planes of surfaces in Minkowski space are nondegenerate.
It is well known that the only nondevelopable minimal ruled surface in
Euclidean 3-space is the classical (right) helicoid and that the only nondevelopable minimal ruled surfaces in Minkowski 3-space are Lorentzian
helicoids (i.e., helicoids of the 1st kind, of the 2nd kind or of the 3rd kind)
and a conjugate of Enneper’s surface of the 2nd kind. All of these surfaces
are helicoidal ruled surfaces (or Schraubregelflächen in German), each of
which is the orbit of a straight line under (the action of) a one-parameter
group of screw motions.
The systematic study of surfaces on which there exists a relationship
between its curvatures was initiated by Julius Weingarten1 and for the
next thirty some years these surfaces were investigated by Beltrami, Darboux and Lie among others. This early research focused primarily on
problems of local nature. More recently, attention has turned to global
properties of Weingarten surfaces, particularly the closed ones. Most of
the investigation (both local and global) concerned surfaces embedded in
Euclidean 3-space.
An old result that was proven independently by Beltrami2 and Dini3
in 1865 states that a nondevelopable ruled surface in Euclidean 3-space
is a Weingarten surface if and only if it is a helicoidal ruled surface. This
result was re-proven by W. Kühnel [20] in 1994 by using another method.
The inner geometry of the second fundamental form has been studied
1
2
A short biography of Julius Weingarten is given in Appendix D.
E. Beltrami, Risoluzione di un problema relativo alla teoria delle superficie gobbe,
Ann. Mat. Pura Appl., (1) 7 (1865), 139–150.
3
U. Dini, Sulle superficie gobbe nelle quali uno dei due raggi di curvatura principali
é una funzione dell’altro, Ann. Mat. Pura Appl., (1) 7 (1865/1866), 205–210.
Introduction
xi
for a very long time. On a nondevelopable surface M , we can regard
the second fundamental form II as a new metric tensor on the semiRiemannian 2-manifold (M, II). As a result, the second Gaussian curvature KII and the second mean curvature HII of (M, II) are defined
formally. It is then worth investigating Weingarten-like surfaces, concerning the existence of a nontrivial functional relation between a pair of
the curvatures K, KII , H and HII . However, these surfaces are still called
Weingarten surfaces.
Let M be a nondevelopable surface on which there exists a nontrivial
functional relation Φ(A, B) = 0, where A, B ∈ {K, KII , H, HII } and A 6=
B. This surface M is termed an {A, B}-W-surface. Kühnel [20] also studied ruled {H, KII }-W-surfaces and ruled {K, KII }-W-surfaces. Recently,
G. Stamou [31] extended Kühnel’s article by classifying ruled {A, B}-
W-surfaces, where (A, B) ∈ {(H, HII ), (K, HII ), (KII , HII )}. Apart from
Weingarten surfaces, Stamou also investigated nondevelopable ruled linear Weingarten surfaces such that the linear combination aKII +bH +cHII
is constant along each ruling, where a, b, c are constants with a2 +b2 +c2 6=
0. Both articles were done in Euclidean 3-space.
In 1999, F. Dillen and W. Kühnel [10] gave a classification of nondevelopable ruled {K, H}-W-surfaces in Minkowski 3-space, paving the way
for the study of other ruled Weingarten surfaces in Minkowski 3-space.
In this dissertation, we complete Dillen and Kühnel’s project by classifying the remaining nondevelopable ruled Weingarten surfaces in Minkowski 3-space; moreover, we also investigate nondevelopable ruled linear
Weingarten surfaces in Minkowski 3-space such that the linear combination aKII + bH + cHII + dK is constant along each ruling, where a, b, c, d
are constants with a2 + b2 + c2 6= 0. The first chapter builds up basic
materials needed in the following chapters. Chapter 2 gives introductions
to vector bundles and some notations in Ricci calculus. In the next two
chapters, we discuss matrix groups and Lorentzian motions. All curva-
xii
Introduction
tures K, KII , H and HII in Minkowski 3-space are defined in Chapter 5.
After that ruled Weingarten surfaces and ruled linear Weingarten surfaces are discussed in Chapters 6 and 7, respectively. Appendices give
extra information that we quote from literature, and some results from
real analysis with detailed proofs, especially the discussion on functional
dependence.
Chapter
1
Preliminaries
This chapter builds up basic materials needed in the following chapters.
We recall basic vector algebra in Minkowski 3-space E31 in Section 1.1
and then give Frenet formulas in Section 1.2. Basic properties of (ruled)
surfaces are discussed in the next two sections. In Section 1.5 shape
operators and principal curvatures are defined. Gauss and Weingarten
equations in E31 are verified in Section 1.6, and some differential operators
are defined in Section 1.7. We close this chapter with the divergence
theorem in semi-Riemannian geometry.
Notations. Let n ∈ Z+ be given.
◮ Rn denotes the vector space (Rn , +, · ). We write each x ∈ Rn in
terms of its components by x = (x1 , . . . , xn ). When we deal with
coordinate neighborhoods of a manifold, we will write the coordinate functions by using superscripts such as (x1 , . . . , xn ) instead of
(x1 , . . . , xn ).
◮ En denotes the Euclidean n-space with Riemannian metric tensor
h· , ·i(n) = dx21 + · · · + dx2n .
1
2
1. Preliminaries
◮ En1 (n > 1) denotes the Minkowski n-space with Lorentzian metric
(n)
tensor h· , ·i1
= −dx21 + dx22 + · · · + dx2n .
(n)
◮ Without ambiguity, we denote both h· , ·i(n) and h· , ·i1
by just
h· , ·i.
1.1
Vector Algebra in E31
In order to get an orthonormal moving frame, we define the cross product
A × B of vectors A and B in E31 by the validity of the equation
hA × B, Ci = (ABC)
for all C ∈ E31 ,
where (ABC) denotes the determinant | A, B, C |. Explicitly, if A =
(a1 , a2 , a3 ) and B = (b1 , b2 , b3 ) belong to E31 , then
hA, Bi = −a1 b1 + a2 b2 + a3 b3 , and
¢
¡
A × B = − (a2 b3 − a3 b2 ), −(a1 b3 − a3 b1 ), a1 b2 − a2 b1 .
The results in the following lemma are similar to the ones in Euclidean
case and can also be found in literature.
Lemma 1.1. Let A, B, C, X, Y, Z ∈ E31 . Then:
(i) A × B = −B × A.
(ii) A × (B + C) = (A × B) + (A × C).
(iii) hA × B, Ai = hA × B, Bi = 0.
(iv) (ABC) = hA, B × Ci = hA × B, Ci.
(v) (A × B) × C = −hA, CiB + hB, CiA.
(vi) (Lagrange identity in E31 )
¯
¯
¯ hA, Xi hB, Xi ¯
¯
¯
hA × B, X × Y i = − ¯
¯.
¯ hA, Y i hB, Y i ¯
1.2 Frenet Formulas in E31
3
¯
¯
¯
¯
hA,
Xi
hB,
Xi
hC,
Xi
¯
¯
¯
¯
¯
(vii) (ABC)(XY Z) = − ¯ hA, Y i hB, Y i hC, Y i ¯¯.
¯
¯
¯ hA, Zi hB, Zi hC, Zi ¯
(viii) A × B = 0 if and only if A and B are linearly dependent.
A vector V in E31 is said to be spacelike if hV, V i > 0 or V = 0, timelike
if hV, V i < 0, and null (or lightlike) if hV, V i = 0 and V 6= 0. We define
p
the norm |V | of V by |V | := |hV, V i| .
1.2
Frenet Formulas in E31
As in Euclidean case, we can define Frenet frames, the curvature and
torsion of a curve in E31 as follows: Let e1 , e2 ∈ E31 be such that hei , ei i =
±1 and he1 , e2 i = 0, and let e3 = e1 ×e2 . Then these three vectors form an
orthonormal frame. If he1 , e1 i = ε and he2 , e2 i = η, where ε, η ∈ {−1, 1},
it follows from the Lagrange identity that he3 , e3 i = −εη. Each vector
X ∈ E31 can be written uniquely in terms of e1 , e2 , e3 by
X = εhX, e1 ie1 + ηhX, e2 ie2 − εηhX, e3 ie3 .
(1.1)
Theorem 1.2 (Frenet formulas in E31 ). ([19], Theorem 2.20). Let c
be a spacelike or timelike curve in E31 such that c is parametrized by arc
length and satisfies hc′′ , c′′ i =
6 0. Then this curve induces a Frenet frame
e1 = c′ ,
c′′
e2 = p
,
|hc′′ , c′′ i|
e3 = e1 × e2 .
Moreover, the following Frenet formulas hold:
  
e′
0
ηκ
 1 
e′  = −εκ
0
 2 
′
e3
0
−ητ
 
e1
 
 
−εητ 
 e2  ,
0
e3
0
4
1. Preliminaries
where ε and η are defined above, and the curvature κ and the torsion
τ of the curve c are defined by
κ = he′1 , e2 i
1.3
and
τ = he′2 , e3 i.
(1.2)
Surfaces in R3
Let Ω be a nonempty open set in R2 .
Definition 1.3. A smooth mapping x : Ω → R3 is called a patch. And
this patch is said to be regular provided that at every point p of Ω the
Jacobian matrix of x at p has rank 2.
When the range of a patch x is contained in a subset M of R3 , we say
that x is a patch in M .
Definition 1.4. Let x : Ω → E3 [resp., x : Ω → E31 ] be a patch. We
define functions E, F, G : Ω → R by
E = hxs , xs i,
F = hxs , xt i,
G = hxt , xt i.
Classically, E, F and G are called the components of the first fundamental
form of the patch x induced from E3 [resp., E31 ].
Suppose further that x is a regular patch with the patch normal U =
xs ×xt
|xs ×xt | .
We define functions e, f, g : Ω → R by
e = −hUs , xs i = hU, xss i,
f = −hUs , xt i = hU, xts i = hU, xst i = −hUt , xs i,
g = −hUt , xt i = hU, xtt i.
Similarly, e, f and g are called the components of the second fundamental
form of x induced from E3 [resp., E31 ].
Definition 1.5. A coordinate patch x : Ω → R3 is a one-to-one regular
patch of an open set Ω in R2 into R3 . A proper patch is a coordinate
patch whose inverse function x−1 : x(Ω) → Ω is continuous.
1.3 Surfaces in R3
5
By a neighborhood of p in M ⊆ R3 we mean a set of all points of M
whose Euclidean distance from p is less than some number ε > 0.
Definition 1.6. A surface M in R3 is a subset of R3 such that for each
point p of M there exists a proper patch in M whose image contains a
neighborhood of p in M .
Note that the image M = x(Ω) of just one proper patch automatically
satisfies Definition 1.6; M is then called a simple surface. (Thus Definition
1.6 says that any surface in R3 can be constructed by gluing together
simple surfaces.)
Definition 1.7. A regular patch x : Ω → R3 whose image lies in a surface
M is called a parametrization of the region x(Ω) in M .
Definition 1.8. Let W be a vector field on R3 and let vp be a tangent
vector to R3 at a point p. Then the covariant derivative of W with respect
to vp is the tangent vector
at the point p.
¯
d ¯¯
∇vp W = ¯ W (p + tvp )
dt t=0
Proposition 1.9. ([26], Lemma 5.2). Let U1 , U2 , U3 be the natural frame
P
field on R3 . If W =
wi Ui is a vector field on R3 and vp is a tangent
vector at p ∈ R3 , then
∇ vp W =
X
vp [wi ]Ui (p).
Proposition 1.10. ([26], p. 81). Let W be a vector field defined on a
region containing a regular curve α. Thus t 7→ W (α(t)) is a vector field
on α called the restriction of W to α and denoted by Wα . Furthermore,
¡ ¢′
∇α′ (t) W = Wα (t).
(1.3)
6
1. Preliminaries
1.4
Ruled Surfaces
A ruled surface M is a surface determined by a parametrization x(s, t) =
α(s) + tβ(s), where α is a curve and β is a nonzero vector field along the
curve α. The curve α is called a base curve and the vector field β is called
a director vector field; we can also regard β as a curve and call it a director
curve. Furthermore, if β × β ′ never vanishes, the ruled surface M is said
to be noncylindrical (with respect to the parametrization x). But, on the
other hand, if β × β ′ vanishes identically, M is called cylindrical (with
respect to x). The rulings of M are the straight lines t 7→ α(s) + tβ(s).
The ruled surface M is said to be conoidal if its rulings are parallel to a
plane.
The following proposition gives sufficient conditions for M to be
conoidal.
Proposition 1.11. Let M : x(s, t) = α(s) + tβ(s) be a ruled surface in
E3 or in E31 . If (β ′′ β ′ β) = 0 and |β| = |β ′ | = 1, then M is conoidal.
Proof. We will discuss only the case where M is in E31 . (The Euclidean
case can be proven similarly.) Suppose that (β ′′ β ′ β) = 0 and |β| = |β ′ | =
1.
Case 1 : hβ, βi = 1. In the moving frame β ′ , β, β ′ × β with signs
ε, 1, −ε (ε = ±1), we have
β ′′ = hβ ′′ , βiβ = −εβ.
Since β is a normal vector field on the unit pseudosphere S12 = {p ∈
E31
:
hp, pi = 1}, we deduce that β is a nonconstant geodesic of S12 ,
obtained by slicing from S12 by a plane passing through the origin of E31 .
This implies that M is conoidal.
Case 2 : hβ, βi = −1. Similarly, we can show that β is a nonconstant
geodesic of the unit pseudohyperbolic space H 2 = {p ∈ E31
: hp, pi = −1},
1.4 Ruled Surfaces
7
obtained by slicing from H 2 by a plane passing through the origin of E31 .
Therefore, M is conoidal.
This completes the proof.
Proposition 1.12. Let M : x(s, t) = α(s) + tβ(s) be a ruled surface in
E3 or in E31 such that hβ, βi is constant and hβ ′ , β ′ i =
6 0. Then there
exists a unique base curve α such that hα′ , β ′ i = 0; moreover, α does not
depend on the choice of the base curve α.
The curve α above is then termed the striction line of M .
Proof. (We can prove this proposition in both Euclidean and Minkowski
cases simultaneously.) Assuming the existence of such a curve α, we get
α(s) = α(s) + u(s)β(s)
(1.4)
for some real-valued function u = u(s), so
α′ = α′ + u′ β + uβ ′ .
Since hβ, β ′ i = 0,
0 = hα′ , β ′ i = hα′ , β ′ i + uhβ ′ , β ′ i.
It follows that u is given by
u=−
hα′ , β ′ i
.
hβ ′ , β ′ i
(1.5)
Thus, if we define α by (1.4) and (1.5), we obtain the required curve. The
“uniqueness” part is obvious.
Suppose that M has another parametrization y(s, v) = γ(s) + vβ(s),
where γ(s) = α(s) + w(s)β(s) for some real-valued function w = w(s).
Let γ be a new base curve of M derived from γ such that hγ ′ , β ′ i = 0.
From the first part,
γ=γ−
hγ ′ , β ′ i
β.
hβ ′ , β ′ i
8
1. Preliminaries
Thus
hα′ + w′ β + wβ ′ , β ′ i
β
hβ ′ , β ′ i
hα′ , β ′ i
=α− ′ ′ β
hβ , β i
γ = α + wβ −
= α.
A ruled surface M : x(s, t) = α(s) + tβ(s) is said to be orthogonal
if α is the striction line and hα′ , βi = 0. The parametrization x(s, t) =
α(s) + tβ(s) of M is said to be of standard form if |β| = |β ′ | = 1 and
hα′ , β ′ i = 0 (therefore, α is automatically the striction line of M ).
Suppose that M : x(s, t) = α(s) + tβ(s) is a ruled surface in E31 . Then
M belongs to a class
•
M11
if the vector fields β and β ′ are nowhere null;
•
M01
if β is nowhere null but β ′ is null everywhere; and
•
M0
if β is null everywhere.
M is said to be
¯
• spacelike if for all p ∈ M, h· , ·ip = h· , ·i¯Tp M is positive definite;
¯
• timelike if for all p ∈ M, h· , ·ip = h· , ·i¯Tp M is nondegenerate of
index 1; and
¯
• lightlike if for all p ∈ M, h· , ·ip = h· , ·i¯Tp M is degenerate.
1.5
Shape Operators
Let M be a surface in E3 or E31 .
1.5 Shape Operators
9
Definition 1.13. Let p be a point of M and let U be a unit normal on
a neighborhood of p in M . Define Sp : Tp M → Tp M by
Sp (vp ) = −∇vp U
for all vp ∈ Tp M . The map Sp is called the shape operator of M at p
derived from U .
At each point p in M there are exactly two shape operators, ±Sp ,
derived from the two unit normals ±U near p. We shall refer to all of
these, collectively, as the shape operator S of M . Thus if a choice of unit
normal is not specified, there is a relatively harmless ambiguity of sign.
Proposition 1.14. ([27], Lemma 4.19). If S is the shape operator of M ,
then at each point p ∈ M ,
S : Tp M → Tp M
is a symmetric linear operator; that is,
hS(vp ), wp i = hvp , S(wp )i
for any pair of tangent vectors vp , wp to M at p.
The eigenvalues (possibly complex) of the shape operator S of M are
called the principal curvatures of M . It is well known that a symmetric linear operator A on a 2-dimensional Lorentzian vector space has a
matrix representation of exactly three types (cf. [24], pp. 166–167):
Ã
!
a1 0
(I) A ∼
,
0 a2
(II) A ∼
(III) A ∼
Ã
!
a 0
and
1 a
Ã
a
b
−b a
!
.
10
1. Preliminaries
Here a, a1 , a2 , b are “real” constants such that b 6= 0. In Cases I and II
the eigenvalues are real, while a ± ib are eigenvalues in Case III.
To illustrate the current situation, we bring here the famous example
of L. K. Graves [15].
B -scroll over a null curve in E31 ). Let α be a null
Example 1.15 (B
curve in E31 with Cartan frame A, B, C; that is, A, B, C are vector fields
along α satisfying the following conditions:
span({A, B, C}) = R3 ,
(ABC) = 1,
hA, Ai = hB, Bi = 0,
hA, Bi = −1,
hA, Ci = hB, Ci = 0,
hC, Ci = 1,
and
α′ (s) = A(s),
A′ (s) = a(s)C(s),
B ′ (s) = bC(s),
C ′ (s) = bA(s) + a(s)B(s),
where b is a constant and a(s) 6= 0 for all s. Then the mapping x : (s, t) 7→
α(s) + tB(s) is a parametrization of a timelike ruled surface M which is
called a B-scroll.
Since M has a unit normal vector field U such that
U (s, t) = btB(s) + C(s),
the shape operator of M derived from U has the matrix representation
[S]B =
Ã
−b
0
−a(s) −b
!
∼
Ã
−b
1
0
−b
!
in the ordered basis B = {xs , xt }. It is easy to check that [S]B is non-
diagonalizable.
1.6 Gauss and Weingarten Equations in E31
1.6
11
Gauss and Weingarten Equations in E31
Conventions.
1. Throughout this dissertation, we use the Einstein summation convention for expressions with indices: if in any term the same index
name appears twice, as both an upper and a lower index, that term
is assumed to be summed over all possible values of that index.
2. With respect to a coordinate system (xi ), the expression Ψ|i stands
for the partial derivative ∂Ψ/∂xi .
Let (M, g) be a semi-Riemannian m-manifold, let (x1 , . . . , xm ) be a
coordinate system on an open set O in M , and let p ∈ O. Then
¯ ª
© ¯
∂1 ¯p , . . . , ∂m ¯p ,
where ∂i := ∂/∂xi (i = 1, . . . , m), forms a basis for the tangent space
Tp M , and
©
¯ ª
¯
dx1 ¯p , . . . , dxm ¯p ,
forms the dual basis for the dual space Tp∗ M . Namely,
¯ ¡ ¯ ¢
dxi ¯p ∂j ¯p = δji
for all i, j ∈ {1, . . . , m}, where δji is the Kronecker delta symbol. On the
open set O, the metric tensor g on M is given by
g = gij dxi dxj .
Here the components gij of g are smooth real-valued functions defined on
O by
gij = g(∂i , ∂j ).
Since g is nondegenerate, the matrix (gij ) is invertible at each point of
O. Let (gij ) denote the inverse matrix of (gij ).
12
1. Preliminaries
We define the connection coefficients Γkij of the Levi-Civita connection
∇ for (xi ) on O by
∇∂i ∂j = Γkij ∂k
(1.6)
for all i, j ∈ {1, . . . , m}. Moreover, each Γkij is given by
ª
1 ©
Γkij = gkl gjl|i + gil|j − gij|l .
2
(1.7)
From now on, we will discuss only surfaces in E31 . On a simple surface
M with parametrization x : Ω → E31 , we know that
Ã
! Ã
!
g11 ◦ x g12 ◦ x
E F
=
.
g21 ◦ x g22 ◦ x
F G
For convenience we will write
Ã
!
g11 ◦ x g12 ◦ x
g21 ◦ x g22 ◦ x
so
Ã
!
g11 g12
simply as
!
Ã
g11 g12
g21 g22
=
Ã
!
E F
F
Similarly, we also drop “◦ x” and write
Ã
Ã
!
G
g11 ◦ x g12 ◦ x
1
=
2
EG − F
−F
g21 ◦ x g22 ◦ x
Γkij ◦ x as
,
g21 g22
G
−F
E
.
!
as
Ã
!
g11 g12
g21 g22
,
Γkij , etc.
The context will indicate what the points of application are.
The equation (1.7) yields very simple expressions of Γkij ’s on a proper
patch in terms of the components E, F, G of the first fundamental form.
Proposition 1.16. ([27], p. 80). Let x : Ω → E31 be a proper patch. Then
GEs − 2F Fs + F Et
,
2(EG − F 2 )
GEt − F Gs
=
,
2(EG − F 2 )
2GFt − GGs − F Gt
=
,
2(EG − F 2 )
2EFs − EEt − F Es
,
2(EG − F 2 )
EGs − F Et
=
,
2(EG − F 2 )
EGt − 2F Ft + F Gs
=
.
2(EG − F 2 )
Γ111 =
Γ211 =
Γ112
Γ212
Γ122
Γ222
1.6 Gauss and Weingarten Equations in E31
13
Corollary 1.17. Let x : Ω → E31 be a proper patch. For each i ∈ {1, 2},
¡ p
¢
Γkik = Γ1i1 + Γ2i2 = ln |EG − F 2 | |i .
Proof. By the Lagrange identity, hxs × xt , xs × xt i = −(EG − F 2 ). Let
U =
xs ×xt
|xs ×xt | ;
then hU, U i = −(EG − F 2 )/|EG − F 2 |. Thus |EG − F 2 | =
−hU, U i(EG − F 2 ) and
Γ111 + Γ212 =
=
=
=
=
Similarly,
Es G + EGs − 2F Fs
2(EG − F 2 )
∂
1
(EG − F 2 )
2
2(EG − F ) ∂s
¢
∂¡
1
− hU, U i(EG − F 2 )
2
−2hU, U i(EG − F ) ∂s
∂
1
|EG − F 2 |
2
2|EG − F | ∂s
¢
∂¡ p
ln |EG − F 2 | .
∂s
Γ121 + Γ222 =
¢
∂¡ p
ln |EG − F 2 | .
∂t
We now establish the well known Gauss and Weingarten equations in
Minkowski 3-space.
Theorem 1.18 (Gauss Equations in E31 ). Let x be a proper patch in
E31 with unit normal vector field U =
xs ×xt
|xs ×xt | .
Then

xss = Γ111 xs + Γ211 xt + ehU, U iU, 



1
2
xst = Γ12 xs + Γ12 xt + f hU, U iU,



x = Γ1 x + Γ2 x + ghU, U iU. 
tt
22 s
(1.8)
22 t
Proof. Since x is regular, it follows that xs , xt and U form a basis for R3
at each point p in the trace of x. Let αi , βi , γi ∈ R (i = 1, 2, 3) be such
14
1. Preliminaries
that
xss

= α1 xs + α2 xt + α3 U, 



xst = β1 xs + β2 xt + β3 U,
xtt = γ1 xs + γ2 xt + γ3 U.
Note that
(1.9)




α3 = ehU, U i, β3 = f hU, U i, γ3 = ghU, U i.
To determine the other coefficients in (1.9), we take the scalar product
of each of the equations in (1.9) with xs and xt . We get:






1

α1 F + α2 G = hxss , xt i = Fs − 2 Et , 





1

β1 E + β2 F = hxst , xs i = Et ,
α1 E + α2 F = hxss , xs i = 21 Es ,
2
(1.10)






1

γ1 E + γ2 F = hxtt , xs i = Ft − 2 Gs , 





1
γ1 F + γ2 G = hxtt , xt i = 2 Gt .
β1 F + β2 G = hxst , xt i =
1
2 Gs ,
The first two equations of (1.10) can be solved for α1 and α2 , yielding
α1 = Γ111 and α2 = Γ211 . The other equations have similar solutions, and
so the proof is complete.
Theorem 1.19 (Weingarten Equations in E31 ). Let x be a proper
patch in E31 with unit normal vector field U =
xs ×xt
|xs ×xt | .
operator S of x is given in terms of the basis {xs , xt } by
f F − eG
eF − f E
xs +
xt
EG − F 2
EG − F 2
f F − gE
gF − f G
xs +
xt .
−S(xt ) = Ut =
2
EG − F
EG − F 2
−S(xs ) = Us =
Then the shape





(1.11)
Proof. Since hU, Us i = hU, Ut i = 0, Us and Ut are tangent vector fields.
Let a, b, c, d ∈ R be such that
Us = axs + bxt
and
Ut = cxs + dxt .
1.7 Some Differential Operators
15
Hence
−e = aE + bF,
−f = cE + dF,
−f = aF + bG,
−g = cF + dG.
We have
a=
f F − eG
,
EG − F 2
b=
eF − f E
,
EG − F 2
c=
gF − f G
,
EG − F 2
d=
f F − gE
.
EG − F 2
This completes the proof.
Remark 1.20. A convenient trick for remembering Weingarten equations
is to think of them as obtained by evaluation of the symbolic determinants
¯
¯
¯
¯
¯
¯
¯
¯
¯ xs xt 0 ¯
¯ xs xt 0 ¯
¯
¯
¯
¯
1
1
¯ E F e ¯ and Ut =
¯ E F f ¯.
Us =
¯
¯
¯
¯
2
2
EG − F ¯
EG − F ¯
¯
¯
¯F G f¯
¯F G g¯
1.7
Some Differential Operators
Let (M, g) be a semi-Riemannian manifold and let F(M ) be the set of all
smooth real-valued functions on M . There are natural generalizations of
the well known differential operators of vector calculus on R3 : gradient,
divergence, and Laplacian.
Definition 1.21. The gradient grad f of a function f ∈ F(M ) is the
vector field metrically equivalent to the differential df ∈ X∗ (M ). Thus
g(grad f, X) = df (X) = Xf
for all X ∈ X(M ).
In terms of a coordinate system df = f|j dxj , hence
grad f = gij f|j ∂i .
(1.12)
In particular, for natural coordinates on semi-Euclidean space we have
16
1. Preliminaries
grad f =
P
εi
∂f
∂i , where εi = g(∂i , ∂i ) for all i.
∂ui
This reduces to the usual formula on R3 .
Definition 1.22. The divergence of a vector field Y on M , denoted by
div Y , is defined as the trace of X 7→ ∇X Y . Thus for a coordinate system
div Y :=
X
∇∂i Y i :=
X
¡
¢
dxi ∇∂i Y = Y|ii + Γiij Y j .
(1.13)
Hence for natural coordinates on semi-Euclidean space,
div Y =
which on R3 is the usual formula.
X ∂Y i
,
∂ui
Proposition 1.23. Let f ∈ F(M ) and let Y be a vector field on M . Then
div f Y = df (Y ) + f div Y.
(1.14)
Proof.
X
¡
¢
dxi ∇∂i (f Y )
X
¡
¢
=
dxi (∂i f )Y + f ∇∂i Y
X¡
¢
=
(∂i f )dxi (Y ) + f dxi (∇∂i Y )
div f Y =
= df (Y ) + f div Y.
Proposition 1.24. ([27], p. 213). Let Y be a vector field on M . For a
coordinate system on M ,
X ∂ ³
1
div Y = q¯
¯
∂xi
¯det I ¯
where det I = det (gij ).
q¯
¯ ´
¯det I ¯ Y i ,
(1.15)
1.8 Divergence Theorem
17
Definition 1.25. The Laplacian ∆f of a function f ∈ F(M ) is the
divergence of its gradient:
∆f = div(grad f ) ∈ F(M ).
Proposition 1.26. ([27], pp. 87 & 213). Let f ∈ F(M ) be given. For a
coordinate system on M , we have
¢
¡
∆f = gij f|i|j − Γkij f|k .
(1.16)
Furthermore, (1.12) and (1.15) imply
1.8
X ∂ ³
1
∆f = q¯
¯
¯det I ¯ ij ∂xi
q¯
´
¯
¯det I ¯ gij ∂f .
∂xj
(1.17)
Divergence Theorem
We conclude this chapter with the divergence theorem in semi-Riemannian
geometry. This section is based on the article of B. Ünal [34].
Let (M, g) be a semi-Riemannian manifold with boundary ∂M , let
p ∈ ∂M , and let ω be a volume form on M .
Definition 1.27. A vector N ∈ Tp M is said to be inward-pointing if
N ∈
/ Tp ∂M and for some ε > 0, there exists a smooth curve segment
α : [0, ε] → M such that α(0) = p and α′ (0) = N . On the other hand,
the vector N is said to be outward-pointing if −N is inward-pointing.
Let V be a finite-dimensional real vector space, let X ∈ V , and let
Ak (V
) be the set of all alternating k-tensor on V . Define a linear map iX :
Ak (V ) → Ak−1 (V ), called interior contraction or interior multiplication
with X, by
iX θ(Y1 , . . . , Yk−1 ) = θ(X, Y1 , . . . , Yk−1 )
for all θ ∈ Ak (V ) and all (Y1 , . . . , Yk−1 ) ∈ V k−1 . In other words, iX θ is
obtained from θ by inserting X into the first slot.
18
1. Preliminaries
Let ∂M+ , ∂M− and ∂M0 denote the sets of all points in ∂M at which
normal vectors to ∂M are spacelike, timelike and null, respectively. Then
∂M ′ := ∂M+ ∪ ∂M− can be considered as the nondegenerate boundary
of M .
Definition 1.28. The induced (semi-Riemannian) volume form of ∂M+
and ∂M− are defined respectively by
ω∂M+ := in+ ω
and
ω∂M− := in− ω,
where n+ , n− are unit outward-pointing normal vector fields on ∂M+ and
∂M− , respectively, and i is the interior contraction.
Notations.
1. Let
n=

n +
n
−
on M+ ,
on M− .
Then n is a unit outward-pointing normal vector field on ∂M ′ .
2. ω∂M ′ := in ω is the volume form on ∂M ′ induced by ω.
Here we state one of the two versions of the divergence theorem by
Ünal.
Theorem 1.29 (Divergence Theorem). ([34], Theorem 3.8). Let
(M, g) be a semi-Riemannian manifold with boundary ∂M , let X be a
vector field on M with compact support, and let n be the unit outwardpointing normal vector field on ∂M ′ . Suppose that ω is a volume form
on M and that ω∂M ′ is the volume form on ∂M ′ induced by ω. If X is
tangent to ∂M at the points of ∂M0 , then
Z
Z
Z
′
g(X, n) ω∂M −
(div X) ω =
M
∂M+
∂M−
g(X, n) ω∂M ′ .
(1.18)
Chapter
2
Ricci Calculus
Ricci calculus is a powerful tool for studying calculus of tensors. We
introduce some notations in Ricci calculus which will be used to define
the second mean curvature HII in Chapter 5.
2.1
Tensors
The notion of tensor field on a manifold generalizes the notions of realvalued function, vector field, and one-form, and thus provides the mathematical means of describing more complicated objects on a manifold.
2.1.1
Basic Algebra
Let V1 , . . . , Vs be modules over a ring R. Then V1 × · · · × Vs is the set
of all s-tuples (v1 , . . . , vs ) with vi ∈ Vi . If Vi = V for all i, the notation
V1 × · · · × Vs is abbreviated to V s . The usual componentwise definitions
of addition and multiplication by an element of R make V1 × · · · × Vs a
module over R, called a direct product (or a direct sum if the notation ×
is replaced by ⊕) of V1 , . . . , Vs . If W is also a module over R, a function
A : V1 × · · · × V s → W
19
20
2. Ricci Calculus
is said to be R-multilinear provided that A is R-linear in each slot, i.e., for
every i ∈ {1, . . . , s}, if vj ∈ Vj for all j 6= i, then the function Ai : Vi → W
defined by
Ai (v) = A(v1 , . . . , vi−1 , v, vi+1 , . . . , vs )
for all v ∈ Vi
is R-linear.
Notation. Let V1 , . . . , Vs , W be modules over a ring R.
Hom(V1 , . . . , Vs ; W ) = {A : V1 × · · · × Vs → W
:
A is R-multilinear}.
Evidently, Hom(V1 , . . . , Vs ; W ) is a module over R under the usual addition of functions and multiplication by an element of R.
Suppose that V is a module over R; let V ∗ be the set of all R-linear
functions from V to R. The usual definition of addition of functions and
multiplication by an element of R make V ∗ a module over R, called the
dual module of V .
By the tensor product U ⊗ V of two modules U and V over R we
mean the module
U ⊗ V = Hom(U ∗ , V ∗ ; R)
= {L : U ∗ × V ∗ → R
:
L is R-bilinear}
over R.
Definition 2.1. For nonnegative integers r, s with r2 + s2 6= 0, an Rmultilinear function
A : V s × (V ∗ )r → R
is called a tensor of type (r, s) over V (or briefly, an (r, s) tensor over V ).
(Note that A : V s → R if r = 0, and A : (V ∗ )r → R if s = 0.) A tensor
of type (0, 0) over V is simply an element of R.
Let Trs (V ) denote the set of all tensors of type (r, s) over V ; it is a
module over R, again with the usual definition of functional addition and
multiplication by an element of R.
2.1 Tensors
2.1.2
21
Tensor Fields
A tensor field A on a manifold M is a tensor over F(M )-module X(M ),
as defined above. Thus if A has type (r, s), it is an F(M )-multilinear
function
A : X(M )s × X∗ (M )r → F(M ).
So A is a multilinear machine which when it is fed r one-forms θ1 , . . . , θr
and s vector fields X1 , . . . , Xs produces a real-valued function
f = A(X1 , . . . , Xs ; θ1 , . . . , θr ) ∈ F(M ).
Here θi occupies the ith contravariant slot and Xj the jth covariant slot
of A.
The set Trs (M ) of all tensor fields on M of type (r, s) is then a module
over F(M ). In the exceptional case r = s = 0, a tensor field on M of type
(0, 0) is just a function f ∈ F(M ); that is, T00 (M ) = F(M ).
While we only add tensors of the same type, any two tensors can be
′
multiplied as follows: If A ∈ Trs (M ) and B ∈ Trs′ (M ), define
′
′
A ⊗ B : X(M )s+s × X∗ (M )r+r → F(M )
by
′
(A ⊗ B)(X1 , . . . , Xs+s′ ; θ1 , . . . , θr+r )
′
= A(X1 , . . . , Xs ; θ1 , . . . , θr )B(Xs+1 , . . . , Xs+s′ ; θr+1 , . . . , θr+r ).
Then A ⊗ B is a tensor of type (r + r′ , s + s′ ), called the tensor product
of A and B. If r′ = s′ = 0, so B is a function f ∈ F(M ); we define
A ⊗ f = f ⊗ A = f A.
Thus if A is also of type (0, 0), the tensor product reduces to ordinary
multiplication in F(M ).
22
2. Ricci Calculus
Evidently the tensor product is F(M )-bilinear, that is,
(f A + gA′ ) ⊗ B = f A ⊗ B + gA′ ⊗ B,
with a similar identity for B. Furthermore, it is immediate from the
definition that the tensor product is associative; thus A ⊗ B ⊗ C is well
defined for tensors of any types. However, the tensor product is generally
not commutative. On the other hand, if A and B are tensor fields on M ,
we have
f (A ⊗ B) = f A ⊗ B = A ⊗ f B
for any f ∈ F(M ).
2.1.3
Interpretations
There are three interpretations that will be used frequently.
(1) If ω is a smooth one-form on a manifold M , then the function
X 7→ ω(X) is F(M )-linear from X(M ) to F(M ), hence it is a (0, 1) tensor
field. Every (0, 1) tensor field arises in this way from a unique one-form,
so we write simply
T01 (M ) = X∗ (M ).
(2) If V is a smooth vector field on M , define V : X∗ (M ) → F(M ) by
V (θ) = θ(V )
for all θ ∈ X∗ (M ).
Obviously, V is F(M )-linear, hence it is a (1, 0) tensor field. Every (1, 0)
tensor field on M arises in this way from a unique vector field, so we write
T10 (M ) = X(M ).
(3) If A : X(M )s → X(M ) is F(M )-linear, we define A : X(M )s ×
X∗ (M ) → F(M ) by
A(X1 , . . . , Xs ; θ) = θ(A(X1 , . . . , Xs ))
2.2 Vector Bundles
23
for all (X1 , . . . , Xs ; θ) ∈ X(M )s × X∗ (M ). Evidently, A is F(M )-multi-
linear, hence it is a (1, s) tensor field. We shall consider A itself to be a
(1, s) tensor field.
The tensors of type (0, s) are said to be covariant, while the tensor of
type (r, 0) with r ≥ 1 are contravariant. For example, smooth real-valued
functions and one-forms are covariant; vector fields are contravariant. An
(r, s) tensor is mixed if neither r nor s is zero. Observe that the definition
of tensor product shows that if A is covariant and B is contravariant, then
A ⊗ B = B ⊗ A.
2.1.4
Tensor Components
Let M be a smooth m-manifold and let ξ = (x1 , . . . , xm ) be a coordinate
system on an open set U ⊆ M . If A ∈ Trs (M ), the components of A
relative to ξ are the real-valued functions
i1
ir
r
Aij11,...,i
,...,js = A(∂j1 , . . . , ∂js ; dx , . . . , dx )
on U , where all indices run from 1 to m.
We have the following lemma:
Lemma 2.2. ([27], Lemma 2.5). Let (x1 , . . . , xm ) be a coordinate system
on U ⊆ M . If A is an (r, s) tensor field, then on U,
A=
X
j1
js
r
Aji11,...,i
,...,js dx ⊗ · · · ⊗ dx ⊗ ∂i1 ⊗ · · · ⊗ ∂ir ,
where each index is summed from 1 to m.
2.2
Vector Bundles
When we glue together the tangent spaces at all points on a manifold M ,
we get a set that can be thought of both as a union of vector spaces and
as a manifold in its own right. This kind of structure is so common in
differential geometry that it has a name.
24
2. Ricci Calculus
Definition 2.3. Let M be a smooth m-manifold. A (smooth) vector
bundle of rank k over M (or a vector bundle over M of fiber dimension k,
or a k-plane vector bundle over M ) is a smooth manifold E of dimension
m + k together with a (surjective) smooth map π : E → M satisfying:
(i) For each p ∈ M , the set Ep := π −1 (p) ⊆ E (called the fiber of E
over p) is endowed with the structure of a k-dimensional real vector
space.
(ii) For each p ∈ M , there exist a neighborhood U of p in M and a
diffeomorphism Φ : π −1 (U) → U × Rk (called a (smooth) local trivialization of E over U), such that the following diagram commutes:
π −1 (U)
FF
FF
F
π FFF
F#
Φ
U
/ U × Rk
xx
xx
x
xx π1
x{ x
(where π1 is the projection on the first factor); and such that for
each q ∈ U, the restriction of Φ to Eq maps the vector space Eq
linearly isomorphically onto the vector space {q} × Rk ∼
= Rk .
We call E the total space, M the base, and π the projection of the
vector bundle. Depending on what we wish to emphasize, we sometimes
omit some or all of the ingredients from the notation, and write “E is a
vector bundle over M ,” or “E → M is a vector bundle,” or “π : E → M
is a vector bundle.” If U is any open set in M , it is easy to verify that
¯
the subset E ¯U := π −1 (U) is again a vector bundle (called the restriction
¯
of E to U) with the restriction of π to E ¯U as its projection map.
There are two familiar examples of vector bundles: the tangent bundle
T M of a smooth manifold M , which is just the disjoint union of the
tangent spaces Tp M for all p ∈ M , and the cotangent bundle T ∗ M ,
which is the disjoint union of the cotangent spaces Tp∗ M := (Tp M )∗ .
Suppose that π : E → M is a vector bundle and U is an open set in
M . A smooth map s : U → E satisfying π ◦ s = IdU is called a (cross)
2.2 Vector Bundles
25
section of E over U. Note that s(p) ∈ Ep for all p ∈ U; thus a section
of E over U is nothing but a correspondence that assigns to each point
¯
p of U the element s(p) in the fiber Ep . Let E(E ¯U ) denote the set of all
¯
sections of E over U. In particular, E(E) := E(E ¯M ). For example, a
section E(T M ) of T M over M is precisely a smooth vector field on M .
Now, let F(U) be the set of all smooth real-valued functions defined on
U. Clearly, F(U) forms a commutative ring under the sum and product
¯
of functions. Let s, s1 , s2 ∈ E(E ¯ ), f, f1 , f2 ∈ F(U) and a, b ∈ R. Define
U
(as1 + bs2 )(p) = as1 (p) + bs2 (p)
¯
¯
for all p ∈ U. Then as1 + bs2 ∈ E(E ¯U ), and E(E ¯U ) becomes a real vector
space under this operation. In addition, we define
(f s)(p) = f (p)s(p)
¯
for all p ∈ U. Clearly, f s ∈ E(E ¯U ). Moreover,
f (s1 + s2 ) = f s1 + f s2 ,
(f1 + f2 )s = f1 s + f2 s,
and
(f1 f2 )s = f1 (f2 s).
¯
Hence E(E ¯U ) is a module over F(U). For instance, the set E(T M ) of all
smooth vector fields over M is a module over F(M ).
Let E and F be vector bundles over the same smooth manifold M .
1. A homomorphism from E into F is a smooth map θ : E → F such
that for each p ∈ M ,
¯
θ¯Ep : Ep → Fp is a linear map.
In particular, a homomorphism θ : E → F which is a diffeomorphism is
called an isomorphism.
2. We can naturally construct new vector bundles such as the dual
bundle E ∗ of E, the tensor product of E and F , and the bundle of homomorphisms Hom(E, F ) from E into F , etc. These new vector bundles
26
2. Ricci Calculus
have the dual space Ep∗ of Ep , the tensor product Ep ⊗ Fp of Ep and Fp ,
and the space of all linear maps Hom(Ep , Fp ) from Ep into Fp as their
fibers over p, respectively.
2.3
Raising and Lowering Indices
Let (M, g) be a semi-Riemannian m-manifold, let (xi ) be a coordinate
system on an open set O in M , and let p ∈ O. The metric tensor g
induces natural linear isomorphisms between Tp M and Tp∗ M ,
♭ : Tp M → Tp∗ M
and
♯ : Tp∗ M → Tp M,
(2.1)
which are the inverses of each other. Indeed, for each Xp ∈ Tp M and
each ωp ∈ Tp∗ M ,
♭(Xp ) := Xp♭ ∈ Tp∗ M
and
♯(ωp ) := ωp♯ ∈ Tp M,
and
gp (ωp♯ , Yp ) = ωp (Yp )
where Xp♭ and ωp♯ satisfy
Xp♭ (Yp ) = gp (Xp , Yp )
(2.2)
for all Yp ∈ Tp M . In the open set O, if we express
¯
Xp = X i (p)∂i ¯p
and
then Xp♭ and ωp♯ are given respectively by
¯
ωp = ωi (p)dxi ¯p ,

¡
¢ ¯
Xp♭ = gij (p)X j (p) dxi ¯p , 
¡
¢ ¯

ωp♯ = gij (p)ωj (p) ∂i ¯p .
(2.3)
It is standard practice to write X ♭ and ω ♯ in coordinates as X ♭ =
Xi dxi and ω ♯ = ω i ∂i , where Xi = gij X j and ω i = gij ωj . One says that
X ♭ [resp., ω ♯ ] is obtained from X [resp., ω] by lowering indices [resp.,
raising indices]. This is why the operations are designated by the musical
notations ♭ = “flat” and ♯ = “sharp.”
2.4 Fiber Metrics
27
By using the above linear isomorphism ♯ , a scalar product g∗p on Tp∗ M
dual to the scalar product gp on Tp M is defined by
¢
¡
g∗p (ωp , θp ) = gp ωp♯ , θp♯
(2.4)
for all ωp , θp ∈ Tp∗ M . From this definition together with (2.3), we get
³¡ ¯ ¢ ¡ ¯ ¢ ´
¯ ¢
¡ ¯
♯
♯
g∗p dxi ¯p , dxj ¯p = gp dxi ¯p , dxj ¯p
¯
¯ ¢
¡
= gp gli (p)∂l ¯p , gkj (p)∂k ¯p
= gli (p)gkj (p)glk (p)
= gij (p).
¡
¢
¡
¢
That is, the inverse matrix gij (p) of gij (p) is precisely the matrix
representing the components of the scalar product g∗p on Tp∗ M .
2.4
Fiber Metrics
Let (M, g) be a semi-Riemannian m-manifold and let p ∈ M . It is easy
to show that the linear space Hom(Tp M ; Tp M ) is linearly isomorphic to
∼ Hom(Tp M, T ∗ M ; R) by a canonical
the tensor product T ∗ M ⊗ Tp M =
p
p
linear isomorphism assigning to each R-linear map f ∈ Hom(Tp M ; Tp M )
an R-bilinear map f † ∈ Hom(Tp M, Tp∗ M ; R) which is defined by
f † (v; ω) = ω(f (v))
for all (v; ω) ∈ Tp M × Tp∗ M .
Let ∇ be the Levi-Civita connection of M . For each Y ∈ E(T M ), we
define
∇Y (X) = ∇X Y
for every X ∈ E(T M ).
We thus obtain a (1, 1) tensor field
∇Y ∈ E(Hom(T M, T M )) ∼
= E(T ∗ M ⊗ T M ).
28
2. Ricci Calculus
Consequently, the Levi-Civita connection ∇ of M defines a map
∇ : E(T M ) → E(T ∗ M ⊗ T M ),
which assigns to each (1, 0) tensor field Y ∈ E(T M ) a (1, 1) tensor field
∇Y ∈ E(T ∗ M ⊗ T M ). We will call ∇Y the covariant differential of Y .
Definition 2.4. Let E → M be a vector bundle. A fiber metric hh· , ·ii
on E is a scalar product on each fiber of E that varies smoothly, in the
sense that for any sections σ, τ ∈ E(E), hhσ(·), τ (·)ii ∈ F(M ).
Example 2.5. g∗ is a fiber metric on T ∗ M .
We shall show that there is a naturally induced connection ∇∗ on
T ∗ M which is compatible with the fiber metric g∗ .
From the linear isomorphisms in (2.1), we get bundle isomorphisms
♭ : T M → T ∗M
and
♯ : T ∗M → T M
between the tangent bundle T M and the cotangent bundle T ∗ M . Let
ω ∈ E(T ∗ M ) and X ∈ E(T M ) be given. We define ∇∗X ω ∈ E(T ∗ M ) by
¡
¢♭
¢
¡
∇∗X ω (Y ) := ∇X ω ♯ (Y )
(2.5)
for all Y ∈ E(T M ), and call ∇∗X ω the covariant derivative of ω by X. It
follows from (2.2) that, for all Y ∈ E(T M ),
¡ ∗ ¢
∇X ω (Y ) = Xω(Y ) − ω(∇X Y ).
(2.6)
Consequently, we may accept (2.6) as a definition of ∇∗X ω. It is easy
to check that ∇∗X ω is F(M )-linear in X and R-linear in ω, and satisfies
Leibniz rule, i.e.,
∇∗X (f ω) = (Xf )ω + f ∇∗X ω
for all f ∈ F(M ).
For each ω ∈ E(T ∗ M ), define
∇∗ ω(X) := ∇∗X ω
2.4 Fiber Metrics
29
for all X ∈ E(T M ). Then we obtain a (0, 2) tensor field
∇∗ ω ∈ E(Hom(T M, T ∗ M )) ∼
= E(T ∗ M ⊗ T ∗ M ).
The map
∇∗ : E(T ∗ M ) → E(T ∗ M ⊗ T ∗ M )
that assigns to each (0, 1) tensor field ω a (0, 2) tensor field ∇∗ ω is called
the connection on T ∗ M induced by ∇. Rewriting (2.6), we have
¢
¡
Xω(Y ) = ∇∗X ω (Y ) + ω(∇X Y ).
This equation explains that the connection ∇ on T M and the connection
∇∗ on T ∗ M are in a mutually dual relationship. Moreover, for all X ∈
E(T M ) and all ω, θ ∈ E(T ∗ M ),








Xg∗ (ω, θ) = Xg(ω ♯ , θ♯ )
= g(∇X ω ♯ , θ♯ ) + g(ω ♯ , ∇X θ♯ )
¡
¢
¡
¢

= g∗ (∇X ω ♯ )♭ , θ + g∗ ω, (∇X θ♯ )♭ 





∗
∗
∗
∗
= g (∇ ω, θ) + g (ω, ∇ θ).
X
(2.7)
X
Namely, ∇∗ is compatible with the fiber metric g∗ on T ∗ M .
It follows from (2.6) that, for all Y = Y j ∂j ∈ E(T M ) and all i, k ∈
{1, . . . , m},
¢
¡
¡
¢
¡ ∗ k¢
∇∂i dx (Y ) = ∂i dxk (Y ) − dxk ∇∂i Y
¢
¡
= ∂i Y k − dxk (∂i Y j )∂j + Y j Γlij ∂l
= ∂i Y k − ∂i Y k − Y j Γkij
= −Γkij dxj (Y ).
This shows that
∇∗∂i dxk = −Γkij dxj
(2.8)
for all i, k ∈ {1, . . . , m}. Comparing (2.8) with (1.6), we note that the
connection coefficients of the connection ∇∗ on T ∗ M induced by ∇ are
quite similar to the ones of ∇ itself but they are of different signs.
30
2. Ricci Calculus
Proposition 2.6. Let γ ∈ F(M ) and let i, j ∈ {1, . . . , m}. Then
¡
¢
∇i γ|j := ∇∗∂i dγ (∂j ) = γ|i|j − Γkij γ|k .
(2.9)
Moreover, ∇j γ|i = ∇i γ|j .
Proof.
¡
¢
∇i γ|j := ∇∗∂i dγ (∂j )
¡
¢
¡
¢
= ∂i dγ(∂j ) − dγ ∇∂i ∂j
¢
¡
= ∂i (γ|j ) − dγ Γkij ∂k
= γ|j|i − Γkij dγ(∂k )
= γ|i|j − Γkij γ|k .
Obviously, ∇j γ|i = ∇i γ|j .
Remark 2.7. Let f ∈ F(M ) be given. It follows from (1.16) that, in
notation of Ricci calculus, the Laplacian of f is
∆f = gij ∇i f|j .
2.5
(2.10)
Parallel Tensor Fields
Let A ∈ E(T ∗ M ⊗T ∗ M ) be a (0, 2) tensor field and let B ∈ E(T M ⊗T M )
be a (2, 0) tensor field. As a generalization of the covariant differentials
∇∗ ω and ∇Y , we can define a tensor field ∇∗ A ∈ E(T ∗ M ⊗ T ∗ M ⊗ T ∗ M )
of type (0, 3) and a tensor field ∇B ∈ E(T ∗ M ⊗ T M ⊗ T M ) of type (2, 1)
respectively by

(∇ A)(X, Y, Z) := XA(Y, Z) − A(∇X Y, Z) − A(Y, ∇X Z), 
∗
(∇B)(X; ω, θ) := XB(ω, θ) − B(∇∗X ω, θ) − B(ω, ∇∗X θ).

(2.11)
We can then define the covariant derivatives ∇∗X A and ∇X B by
¡ ∗ ¢
¡
¢
∇X A (Y, Z) := ∇∗ A(X, Y, Z) and ∇X B (ω, θ) := ∇B(X; ω, θ).
2.6 The Ricci Lemma
31
As usual, ∇∗X A [resp., ∇X B] is F(M )-linear in X and R-linear in A [resp.,
R-linear in B], and satisfies Leibniz rule
∇∗X (f A) = (Xf )A + f ∇∗X A
£
¤
resp., ∇X (f B) = (Xf )B + f ∇X B .
If ∇∗ A ≡ 0 [resp., ∇B ≡ 0], then A [resp., B] are called parallel
tensor fields with respect to the connections ∇∗ [resp., ∇]. It follows
from (2.11) and (2.7) that
∇∗ g ≡ 0
∇g∗ ≡ 0.
Namely, g is a parallel tensor field of type (0, 2) with respect to ∇∗ , and
g∗ is a parallel tensor field of type (2, 0) with respect to ∇.
2.6
The Ricci Lemma
Let (xi ) be a local coordinate system in a local coordinate neighborhood
O of M . In Ricci calculus, we express the components of the (0, 3) tensor
field ∇∗ g and the (2, 1) tensor field ∇g∗ by ∇i gjk and ∇i gjk , respectively.
More explicitly, ∇i gjk and ∇i gjk are smooth functions in O such that
¢
¢ ¡
¢¡
¢¡
¡
∇i gjk = ∇∗∂i g ∂j , ∂k = ∇∗ g ∂i , ∂j , ∂k ,
¢
¢¡
¢ ¡
¢¡
¡
∇i gjk = ∇∂i g∗ dxj , dxk = ∇ g∗ ∂i ; dxj , dxk .
From (1.6), (2.4), (2.8) and (2.11), simple computations yield that

l
l
∇i gjk = gjk|i − Γij glk − Γik gjl , 
∇i gjk = gjk |i + Γjil glk + Γkil gjl . 
(2.12)
Consequently, we may regard (2.12) as definitions of ∇i gjk and ∇i gjk .
That g and g∗ are parallel with respect to ∇∗ and ∇, respectively, means
∇i gjk ≡ 0
and
∇i gjk ≡ 0
(2.13)
for all i, j, k ∈ {1, . . . , m}. These two equations in (2.13) are known as
the Ricci lemma.
32
2. Ricci Calculus
2.7
Contraction
There is a remarkable operation called contraction that shrinks (r, s) tensors to (r − 1, s − 1) tensors. The general definition derives from the
following special case.
Lemma 2.8. ([27], Lemma 2.6). Let M be a smooth manifold. Then
there is a unique F(M )-linear function C : T11 (M ) → F(M ), called (1, 1)
contraction, such that C(θ ⊗ X) = θX for all X ∈ X(M ) and all θ ∈
X∗ (M ).
Proof. Since C is to be F(M )-linear, it will be a pointwise operation. On
a coordinate neighborhood U, a (1, 1) tensor field A can be written as
P i j
Aj dx ⊗ ∂i . Since C(dxj ⊗ ∂i ) must be dxj (∂i ) = δij , this inspires us
to define
C(A) =
X
Aii =
X
A(∂i ; dxi ).
Then C has the required properties on U. To obtain the required global
function it suffices to show that this definition is independent of the choice
of coordinate system. Let (y i ) be another coordinate system on U. Thus
X
k
A
´ X ³X ∂xi ∂ X ∂y k
´
³ ∂
k
j
=
A
;
dy
;
dx
∂y k
∂y k ∂xi
∂xj
k
i
j
´
X ∂xi ∂y k ³ ∂
j
=
A
;
dx
∂xi
∂y k ∂xj
i,j,k
³ ∂
´
X
j
=
δji A
;
dx
∂xi
i,j
´
X ³ ∂
i
=
A
.
;
dx
∂xi
i
To extend (1, 1) contraction C to tensors of higher type the scheme is
to specify one covariant slot and one contravariant slot, and apply C to
these.
2.7 Contraction
33
Suppose A ∈ Trs (M ) and 1 ≤ i ≤ r and 1 ≤ j ≤ s. Fix one-forms
θ1 , . . . , θr−1 and vector fields X1 , . . . , Xs−1 . Then the function
(X; θ) → A(X1 , . . . , Xj−1 , X, Xj , . . . , Xs−1 ; θ1 , . . . , θi−1 , θ, θi . . . , θr−1 ).
is a (1, 1) tensor field that can be written as
A(X1 , . . . , Xj−1 , · , Xj , . . . , Xs−1 ; θ1 , . . . , θi−1 , · , θi . . . , θr−1 ).
Applying the (1, 1) contraction to this tensor field produces a real-valued
function denoted by (Cji A)(X1 , . . . , Xs−1 ; θ1 , . . . , θr−1 ); thus
(Cji A)(X1 , . . . , Xs−1 ; θ1 , . . . , θr−1 )
X
=
A(X1 , . . . , Xj−1 , ∂k , Xj , . . . , Xs−1 ; θ1 , . . . , θi−1 , dxk , θi . . . , θr−1 ).
k
Evidently, Cji A is F(M )-multilinear in its arguments. Hence it is a tensor
field of type (r − 1, s − 1) called the contraction of A over i, j.
Example 2.9. Let (M, g) be a semi-Riemannian manifold. Define a (1, 1)
tensor field δ : X(M ) × X∗ (M ) → F(M ) on M by
δ(X; θ) := θX
for all θ ∈ X∗ (M ) and all X ∈ X(M ). Observe that
δ = C21 (g ⊗ g∗ )
since, relative to a coordinate system,
¡
¢i
δ ij = δji = gjk gki = C21 (g ⊗ g∗ ) j .
In addition, we define
∇k δ ij := (∇k δ)(∂j ; dxi )
:= δ ij|k − δ(∂j ; ∇∗k dxi ) − δ(∇k ∂j ; dxi ),
so
i
∇k δ ij = δj|k
+ Γikl δjl − Γlkj δli = 0.
(2.14)
Chapter
3
Matrix Groups
This chapter is devoted to the notion of matrix groups which we will need
to classify all nontrivial one-parameter groups of Lorentzian motions in
E31 in the next chapter.
Throughout this chapter, k will denote the real number field R or the
complex number field C.
3.1
Groups of Matrices
Let Mm,n (k) be the set of m × n matrices whose entries are in k. We will
denote the (i, j) entry of an m × n matrix A by Aij or aij , and also write

a11
 .
.
A = (aij ) = 
 .
···
..
.
am1 · · ·
We will use the special notations
Mn (k) := Mn,n (k),

a1n
.. 
. 
.
amn
kn := Mn,1 (k).
Mm,n (k) is a k-vector space with the operations of matrix addition
and scalar multiplication. For all r ∈ {1, . . . , m} and all s ∈ {1, . . . , n},
35
36
3. Matrix Groups
the matrices E rs with

1 if i = r and j = s,
¡ rs ¢
E ij = δir δjs =
0 otherwise,
form a basis for Mm,n (k). Hence the dimension of Mm,n (k) is equal to mn.
We will denote the standard basis vectors of kn = Mn,1 (k) by e1 , . . . , en ,
where
er = E r1
for each r ∈ {1, . . . , n}.
Notations.
1. GLn (k) = {A ∈ Mn (k)
:
det A 6= 0} is the set of all invertible
2. SLn (k) = {A ∈ Mn (k)
:
det A = 1} ⊆ GLn (k) is the set of all
n × n matrices,
and
n × n unimodular matrices.
Theorem 3.1. ([1], Theorem 1.2). The set GLn (k), SLn (k) are groups
under matrix multiplication. Furthermore, SLn (k) is a subgroup of GLn (k)
or in notation SLn (k) 6 GLn (k).
Because of this group structures, GLn (k) is called the n × n general
linear group, while SLn (k) is called the n × n special linear group or n × n
unimodular group.
3.2
Groups of Matrices as Metric Spaces
In order to consider the set Mn (k) as a metric space, we will define a
norm k · k on Mn (k). We begin with the usual notion of length for a
vector x ∈ kn , namely
|x| =
p
|x1 |2 + · · · + |xn |2 .
This is an example of a norm on the vector space kn .
3.2 Groups of Matrices as Metric Spaces
For A ∈ Mn (k) we define
kAk = sup
½
|Ax|
|x|
:
n
37
¾
x ∈ k r {0} .
Note that |Ax| ≤ kAk|x| for all x ∈ kn and that
kAk = sup{|Ax|
:
x ∈ kn and |x| = 1}.
For a real matrix A ∈ Mn (R) ⊆ Mn (C), at the first sight
kAkR = sup{|Ax|
:
x ∈ Rn and |x| = 1}
kAkC = sup{|Ax|
:
x ∈ Cn and |x| = 1}
and
seem to be different. Actually, they are identical.
Lemma 3.2. ([1], Lemma 1.3). If A ∈ Mn (R), then kAkC = kAkR .
The main properties of k · k are summarized in the next result.
Proposition 3.3. ([1], Proposition 1.5). The function k · k : Mn (k) → R
has the following properties:
(i) If t ∈ k and A ∈ Mn (k), then ktAk = |t|kAk.
(ii) If A, B ∈ Mn (k), then kABk ≤ kAkkBk.
(iii) If A, B ∈ Mn (k), then kA + Bk ≤ kAk + kBk.
(iv) If A ∈ Mn (k), then kAk = 0 if and only if A = 0.
(v) kIn k = 1, where In ∈ Mn (k) is the identity matrix.
Define ρ : Mn (k) × Mn (k) → R by ρ(A, B) = kA − Bk for all A, B ∈
Mn (k). Clearly, ρ is a metric on M . Without ambiguity we say that
(Mn (k), k · k) is a metric space.
38
3. Matrix Groups
An open disc in Mn (k) with center A ∈ Mn (k) and radius r > 0,
denoted by NMn (k) (A; r), is the set
{B ∈ Mn (k)
:
kB − Ak < r}.
Similarly, if Y ⊆ Mn (k) and A ∈ Y , we set
NY (A; r) = {B ∈ Y
:
kB − Ak < r} = NMn (k) (A; r) ∩ Y.
It can be shown that the metric topologies induced by k · k and the
2
usual norm on kn agree in the sense that they consist of the same open
sets.
Proposition 3.4. ([1], Proposition 1.7). For every r, s ∈ {1, . . . , n}, the
coordinate function coordrs : Mn (k) → k, defined by coordrs (A) = Ars ,
is continuous.
2
Corollary 3.5. ([1], Corollary 1.8). If f : kn → k is continuous, then the
¢
¡
associated function F : Mn (k) → k, defined by F (A) = f (Aij )1≤i,j≤n ,
is continuous.
Corollary 3.6. ([1], Corollary 1.9). The determinant det : Mn (k) → k
and trace tr : Mn (k) → k are continuous functions.
Proposition 3.7. ([1], Proposition 1.13). A function F : Mm (k) →
Mn (k) is continuous with respect to the norms k · k if and only if each of
the component functions Frs : Mm (k) → k is continuous. In particular,
a function f : Mm (k) → k is continuous with respect to the norm and
the usual metric on k if and only if it is continuous when viewed as a
2
function from km to k.
Proposition 3.8. ([1], Proposition 1.14).
(i) GLn (k) is an open subset of Mn (k).
(ii) SLn (k) is a closed subset of Mn (k).
Proposition 3.9. ([1], p. 11). The matrix addition, matrix multiplication
and matrix inverse operator are continuous.
3.3 Matrix Groups
3.3
39
Matrix Groups
Definition 3.10. A subgroup G of GLn (k) is called a matrix group over
k or a k-matrix group if it is a closed subset of GLn (k). For clarity we
sometimes say that G is a matrix subgroup of GLn (k).
Example 3.11. SLn (k) is a matrix subgroup of GLn (k).
The next result follows immediately from the properties of topological
subspaces.
Proposition 3.12. ([1], Proposition 1.31). Let G be a matrix subgroup
of GLn (k). Then a subgroup H of G which is also closed in G is a matrix
subgroup of GLn (k).
This proposition suggests another definition.
Definition 3.13. A closed subgroup of a matrix group G is called a
matrix subgroup of G.
Proposition 3.14. ([1], Proposition 1.33). If K is a matrix subgroup of
a matrix group G and if H is a matrix subgroup of K, then H is a matrix
subgroup of G.
3.4
Some Important Examples
We now discuss some important examples of real and complex matrix
groups.
Affine Groups
The n-dimensional affine group over k is
(Ã
!
)
A t
n
Aff n (k) =
: A ∈ GLn(k) and t ∈ k 6 GLn+1(k).
0 1
40
3. Matrix Groups
This is clearly a matrix subgroup of GLn+1 (k). If we identify x ∈ kn with
( x1 ) ∈ kn+1 , then
Ã
!Ã !
A t
x
0
1
1
=
Ã
!
Ax + t
1
.
The transformations of kn with the form x 7→ Ax + t, where A is an
invertible matrix, are called affine transformations.
The vector space kn itself can be viewed as the translation subgroup
Transn (k) of Aff n (k), where
(Ã
!
In t
Transn (k) =
0 1
:
t ∈ kn
)
6 Aff n (k),
which is obviously a matrix subgroup of Aff n (k).
Example 3.15. We may consider GLn (k) as a matrix subgroup of Aff n (k)
¡
¢
0
by identifying n × n matrices A with A
0 1 . We can restrict to this embedding of SLn (k) which then appears as a closed subgroup of SLn+1 (k) 6
GLn+1 (k). So SLn (k) is a matrix subgroup of SLn+1 (k).
More generally, with the aid of the above embedding, any matrix
subgroup of GLn (k) can also be viewed as a matrix subgroup of GLn+1 (k)
and of Aff n (k).
The following is a standard notion in group theory.
Definition 3.16. Let G be a group, H a subgroup of G and N a normal
subgroup of G. Then G is the semidirect product of H and N if G =
HN (= N H) and H ∩ N = {e}, where e is the identity element of G.
Notations.
1. If N is a normal subgroup of G, we write N ⊳ G.
2. If H 6 G, N ⊳ G and G is the semidirect product of H and N , we
write G = H ⋉ N or G = N ⋊ H. Nevertheless, we will reserve the
notation G = H ⋉ N [resp., G = N ⋊ H] and use it whenever we
regard G as HN [resp., N H].
3.4 Some Important Examples
41
Remark 3.17. If G = H ⋉ N [resp., G = N ⋊ H], then for each g ∈ G,
there are unique h ∈ H and a ∈ N such that g = ha [resp., g = ah].
Proposition 3.18. ([1], Proposition 1.37).
(i) Transn (k) ⊳ Aff n (k).
(ii) Aff n (k) = Transn (k) ⋊ GLn (k)
= {T A
:
T ∈ Transn (k), A ∈ GLn (k)}.
Orthogonal Groups
A matrix A ∈ Mn (k) is called an orthogonal matrix if AT A = In , where
AT denotes the transpose of A. Let
O(n) = {A ∈ GLn (R)
:
AT A = In };
then O(n) is a subgroup of GLn (R) and is called the n×n (real ) orthogonal
group. The single matrix equation AT A = In is equivalent to the n2
equations
n
X
aki akj = δji
k=1
(i, j ∈ {1, . . . , n}).
This means that O(n) is a closed subset of Mn (k) and so is a matrix
subgroup of GLn (R). For A ∈ O(n),
(det A)2 = det(AT A) = det In = 1,
which implies that det A = ±1. Thus we have
O(n) = O+ (n) ∪ O− (n),
where
O+ (n) = {A ∈ O(n)
:
det A = 1}
O− (n) = {A ∈ O(n)
:
det A = −1}.
and
42
3. Matrix Groups
Moreover, O+ (n) is a matrix subgroup of GLn (R); we call this subgroup
the n × n special orthogonal group and denote it by SO(n).
3.5
Lorentzian Groups
In this section, we discuss some matrix groups of isometries.
Definition 3.19. Let (M, gM ) and (N, gN ) be semi-Riemannian manifolds. An isometry from M onto N is a diffeomorphism φ : M → N that
preserves metric tensors, i.e., φ∗ (gN ) = gM . In other words,
¡
¢
gN φ∗ (vp ), φ∗ (wp ) = gM (vp , wp )
for all p ∈ M and all vp , wp ∈ Tp M . If φ : M → M is an isometry from
M onto M itself, we say that φ is an isometry in M .
Let n ∈ Z be such that n ≥ 2.
Now we define a Lorentzian scalar product on Rn in terms of matrices.
With the nonsingular diagonal matrix
Qn = diag(−1, 1, . . . , 1 ) ∈ Mn (R) ,
| {z }
n−1 terms
we define a Lorentzian scalar product h· , ·i on Rn by
hx, yi = xT Qn y = −x1 y1 + x2 y2 + · · · + xn yn .
It is standard to denote this scalar product space by En1 and call it the
Minkowski space of dimension n (or shortly, the Minkowski n-space). We
will also use t in place of x1 since in Relativity this coordinate is related
to the time measurement while the others are related to spatial ones. We
will often write elements of En1 in the form
X=
à !
t
x
,
where x ∈ Rn−1 and t ∈ R.
3.5 Lorentzian Groups
Thus, if Xi =
à !
ti
43
∈ En1 (i = 1, 2), then hX1 , X2 i = −t1 t2 +hx1 , x2 i(n−1) .
xi
The set of all linear isometries in En1 is the same as the set O1 (n) of
all matrices A ∈ GLn (R) that preserves the scalar product h· , ·i. The
following lemma gives criteria for matrices A ∈ Mn (R) to be in O1 (n).
Lemma 3.20. ([27], p. 234). The following conditions on a matrix A ∈
Mn (R) are equivalent:
(i) A ∈ O1 (n).
(ii) AT Qn A = Qn .
(iii) AT ∈ O1 (n).
(iv) The columns [resp., rows] of A form an orthonormal basis for En1
such that the first column [resp., row] is timelike.
(v) A carries one (hence every) orthonormal basis for En1 to an orthonormal basis.
Remarks 3.21.
1. O1 (n) is a matrix subgroup of GLn (R). This group is called the
Lorentzian group of all linear isometries in Minkowski n-space En1 .
The elements of O1 (n) are called linear Lorentzian rotations (or
linear Lorentzian transformations). The matrix group O1 (n) is also
termed a linear Lorentzian rotation group.
2. For all A ∈ O1 (n), det A = ±1.
The nullcone, denoted by Hn (0), is the set of all null vectors in En1 ,
i.e.,
Hn (0) = {( xt ) ∈ En1
:
( xt ) 6= 0 and h( xt ) , ( xt )i = hx, xi(n−1) − t2 = 0}.
44
3. Matrix Groups
Note that Hn (0) = Hn+ (0) ∪ Hn− (0), where
Hn+ (0) = {( xt ) ∈ Hn (0)
:
t > 0} and Hn− (0) = {( xt ) ∈ Hn (0)
:
t < 0}.
For each positive real number r, let
Hn (r) = {( xt ) ∈ En1
:
h( xt ) , ( xt )i = hx, xi(n−1) − t2 = −r};
then Hn (r) = Hn+ (r) ∪ Hn− (r), where
Hn+ (r) = {( xt ) ∈ Hn (r)
:
t > 0} and Hn− (r) = {( xt ) ∈ Hn (r)
:
t < 0}.
There is a relation between these two sets.
Lemma 3.22. For every real number r ≥ 0,
−Hn+ (r) = {−X
−Hn− (r) = {−X
:
:
X ∈ Hn+ (r)} = Hn− (r)
and
X ∈ Hn− (r)} = Hn+ (r).
Lemma 3.23. For each real number r ≥ 0 and each A ∈ O1 (n),
AHn+ (r) = Hn+ (r) if and only if AHn− (r) = Hn− (r).
Proof. By Lemma 3.22, we have
AHn+ (r) = Hn+ (r) ⇐⇒ A(−Hn− (r)) = −Hn− (r)
⇐⇒ −AHn− (r) = −Hn− (r)
⇐⇒ AHn− (r) = Hn− (r).
It is standard to put
SO1 (n) = {A ∈ O1 (n)
:
det A = 1} = O1 (n) ∩ SLn (R) 6 O1 (n).
Thus SO1 (n) is a matrix subgroup of O1 (n) which is termed a proper
Lorentzian group.
3.5 Lorentzian Groups
45
Definition 3.24. A Lorentzian rotation in En1 around an axis ℓ is an
isometry in En1 which leaves the axis ℓ pointwise fixed. We call ℓ the axis
of rotation.
Lemma 3.25. Suppose that n is an odd integer and A ∈ SO1 (n). Then:
(i) A has an eigenvalue 1.
(ii) If v is an eigenvector corresponding to the eigenvalue 1, then A
leaves the axis ℓ = span({v}) pointwise fixed.
As a result, A is a linear Lorentzian rotation around the axis
of rotation ℓ and SO1 (n) is a group of linear Lorentzian rotations
around the axes of rotation passing through the origin of En1 .
Proof. (i) Since
¢
¢
¡
¡
det A − In = det AT − In
¡
¢
= det AT − (AT Qn AQn ) (∵ AT Qn AQn = Qn Qn = In )
¡
¢
= det In − (Qn AQn )
(∵ det AT = 1)
¡
¢
= det (Qn Qn ) − (Qn AQn )
¡
¢
= det In − A
¢
¡
= − det A − In ,
we deduce that det(A − In ) = 0, so A has an eigenvalue 1.
(ii) Clear.
Note that some elements of SO1 (n) interchange the sets Hn± (1). We
thus define a group Lor(n) to be a matrix subgroup of SO1 (n) whose
elements preserve each of the sets Hn± (1). It follows from Lemma 3.23
that
Lor(n) = {A ∈ SO1 (n)
:
AHn+ (1) = Hn+ (1)} 6 SO1 (n)
Lor(n) = {A ∈ SO1 (n)
:
AHn− (1) = Hn− (1)} 6 SO1 (n).
or
46
3. Matrix Groups
Proposition 3.26. ([1], Proposition 6.3). For each r ≥ 0,
Lor(n) = {A ∈ SO1 (n)
:
AHn± (r) = Hn± (r)} 6 SO1 (n).
Proposition 3.27. For each r ≥ 0,
Lor(n) = {A ∈ SO1 (n)
:
AHn± (r) ⊆ Hn± (r) and A−1 Hn± (r) ⊆ Hn± (r)}.
Note that, for all r > 0, Hn± (r) are path-connected components of
Hn (r). The following shortcut to determine Lor(n) is thus available.
Corollary 3.28. Lor(n) = {(aij ) ∈ SO1 (n)
:
a11 > 0}.
Proof. Since Ae1 ∈ Hn+ (1) for all A ∈ Lor(n), we deduce that
Lor(n) ⊆ {(aij ) ∈ SO1 (n)
Conversely, let A ∈ {(aij ) ∈ SO1 (n)
:
Hn+ (1)
:
a11 > 0}.
a11 > 0} be arbitrary. Since e1 ∈
and Ae1 ∈ Hn+ (1), it follows that AHn+ (1) ⊆ Hn+ (1). Similarly,
we can show that AHn− (1) ⊆ Hn− (1). Note that A−1 Hn+ (1) ⊆ Hn+ (1) or
A−1 Hn+ (1) ⊆ Hn− (1). If A−1 Hn+ (1) ⊆ Hn− (1), then Hn+ (1) ⊆ AHn− (1) ⊆
Hn− (1), which is a contradiction. Hence A−1 Hn+ (1) ⊆ Hn+ (1). By Proposition 3.27, A ∈ Lor(n).
This completes the proof.
Notes.
1. We call the matrix group
Lor(n) = {(aij ) ∈ SO1 (n)
:
a11 > 0}
a proper orthochronous Lorentzian group and denote it by
Lorn (+, ↑).
2. In fact, Lorn (+, ↑) is the connected component of the identity in
O1 (n). (See [27], pp. 237–238.)
3.6 One-parameter Groups in Matrix Groups
3.6
47
One-parameter Groups in Matrix Groups
Definition 3.29. Let ] a, b [ be an open interval in R and let t ∈ ] a, b [ .
A function γ : ] a, b [ → Mn (k) is said to be differentiable at t if the limit
lim
s→t
γ(s) − γ(t)
s−t
exists as an element of Mn (k). We call this limit the derivative of γ at t,
denoted by γ ′ (t); thus
γ ′ (t) = lim
s→t
γ(s) − γ(t)
∈ Mn (k).
s−t
It is easy to check that γ : ] a, b [ → Mn (k) is differentiable at t ∈ ] a, b [
if and only if each coordinate function γij (s) := (γ(s))ij is differentiable
at t. Moreover,
′
γ ′ (t) = (γij
(t)).
Definition 3.30. Let G be a matrix subgroup of GLn (k). A continuous
function γ : ] a, b [ → G is termed a curve in G. If the derivative γ ′ (t)
exists at each t ∈ ] a, b [ , then γ is a differentiable curve in G.
Notes 3.31.
1. All matrix subgroups of GLn (k) are Lie groups, so they are smooth
manifolds.
2. The curve γ in Definition 3.30 is said to be smooth if it is considered
as a smooth mapping between manifolds.
3. The curve γ is smooth if and only if its derivatives of all order exist
at each t ∈ ] a, b [ . Equivalently, the curve γ is smooth if and only
if its coordinate functions γij are smooth real-valued functions on
] a, b [ .
Definition 3.32. Let G be a matrix subgroup of GLn (k). A one-parameter
group in G is a smooth curve γ : R → G and also satisfies
48
3. Matrix Groups
γ(s + t) = γ(s) · γ(t)
for all s, t ∈ R.
Remark. For a one-parameter group in G 6 GLn (k), γ(0) = In .
Example 3.33. Let γ : R → GL3 (R) be defined by


1
0
0



γ(s) = 
0 cos s − sin s
0 sin s cos s
for all s ∈ R. Then γ is a one-parameter group in GL3 (R).
We close this chapter with the expressions of one-parameter groups
relating to the matrix exponential exp.
Theorem 3.34. ([1], Theorem 2.17). Let γ : R → G be a one-parameter
group in a matrix group G. Then for all s ∈ R,
γ(s) = exp(sA) ,
where A = γ ′ (0) and exp(sA) = In + sA +
(3.1)
(sA)2 (sA)3
+
+ ··· .
2!
3!
Chapter
4
Lorentzian Motions in E31
In this chapter, we will give an elementary approach to classify all nontrivial one-parameter groups of Lorentzian motions in E31 . By trivial cases
we mean pure translation groups.
4.1
Groups of Linear Lorentzian Rotations
In this section, we will confine ourselves to the matrix group SO1 (3).
According to Lemma 3.25, a linear Lorentzian rotation in SO1 (3) is indeed
a linear Lorentzian rotation around an axis passing through the origin.
Depending on the axis of rotation being timelike, spacelike or null, a
Lorentzian rotation can be represented by one of the following matrices
up to conjugation in the Lorentzian group O1 (3):
 
 

2
1 + s2
cosh s sinh s 0
1
0
0
 
 

0 cos s − sin s ,  sinh s cosh s 0 ,  s2
  2
 

s
0
0
1
0 sin s cos s
2
− s2
s
−s
1
1−
s2
2


s
.
The first two matrices have an eigenvalue 1 with multiplicity 1 or 3,
while the third matrix has a unique eigenvalue 1 with multiplicity 3.
Furthermore, the second matrix also has eigenvalues cosh s ± sinh s, but
49
4. Lorentzian Motions in E31
50
the first one does not. As a result, these 3 matrices are, in general, not
similar to one another.
Before investigating Lorentzian rotations, we need the following lemma
which gives smooth angle functions.
Let F(R) denote the set of all smooth real-valued functions defined on
R.
Lemma 4.1 (Angle functions). ([26], p. 50). Let f, g ∈ F(R) be given.
Suppose that f 2 + g 2 = 1 and that ϕ0 is a number such that f (0) = cos ϕ0
and g(0) = sin ϕ0 . If ϕ : R → R is a function defined by
Z s
ϕ(s) = ϕ0 +
(f g ′ − gf ′ )
0
for all s ∈ R, then ϕ ∈ F(R); moreover,
f = cos ϕ
and
g = sin ϕ.
Let γ : R → Lor3 (+, ↑) be a one-parameter group of Lorentzian rota-
tions and let A ∈ M3 (F(R)) be such that

a1 (s) a2 (s)


γ(s) = A(s) =  b1 (s) b2 (s)
c1 (s) c2 (s)
Remark.
det A(s) = 1 for all s ∈ R.
for all s ∈ R,

a3 (s)

b3 (s) 
 ∈ Lor3 (+, ↑).
c3 (s)
The matrix A is determined by the type of the axis of rotation.
Linear Lorentzian Rotations around a Timelike Axis
Suppose that the axis of rotation is the timelike axis ℓ = {(t, 0, 0)
:
t∈
R}. Since Ae1 = e1 and since hAe1 , Aej i = he1 , ej i = 0 for all j ∈ {2, 3},
we obtain that


1 0 0



A=
0 b2 b3  .
0 c 2 c3
4.1 Groups of Linear Lorentzian Rotations
51
Note that hAei , Aej i = hei , ej i = δji for all i, j ∈ {2, 3}, so
b22 + c22 = 1,
(4.1)
b23 + c23 = 1,
(4.2)
b2 b3 + c2 c3 = 0,
and
(4.3)
b2 c3 − b3 c2 = 1.
(4.4)
Hence b2 = c3 and b3 = −c2 . Since (b2 (0), c2 (0)) lies on the unit circle
{(x, y)
:
x2 + y 2 = 1}, b2 (0) = cos ϕ0 and c2 (0) = sin ϕ0
for some
ϕ0 ∈ R. It follows from Lemma 4.1 that there is a ϕ ∈ F(R) such that
for all s ∈ R,


1
0
0



A(s) = 
0 cos ϕ(s) − sin ϕ(s) .
0 sin ϕ(s) cos ϕ(s)
Since A(s + t) = A(s) · A(t) for all s, t ∈ R, there exists a function
n : R × R → Z be such that ϕ(s) + ϕ(t) = ϕ(s + t) + 2πn(s, t) for all
s, t ∈ R. Obviously, n is constant and n ≡ ϕ(0)/(2π). Thus ϕ(s + t) =
ϕ(s)+ϕ(t)−ϕ(0) for all s, t ∈ R. Define ϕ
e : R → R by ϕ(s)
e = ϕ(s)−ϕ(0)
for all s ∈ R. Clearly, ϕ
e is a smooth additive function, i.e., ϕ(s
e + t) =
ϕ(s)
e + ϕ(t)
e
for all s, t ∈ R. It follows from Proposition A.2 that there
exists a ∈ R such that ϕ(s)
e = as for all s ∈ R. Thus ϕ(s) = as + ϕ(0) for
all s ∈ R. But ϕ(0) = 2πn, so

1
0
0




A(s) = 
0
cos(as
+
ϕ(0))
−
sin(as
+
ϕ(0))



0 sin(as + ϕ(0))
cos(as + ϕ(0))

1
0
0



= 0 cos as − sin as

0 sin as cos as
4. Lorentzian Motions in E31
52
for all s ∈ R. If a 6= 0, we can change the variable s so that


1
0
0



A(s) = 
0 cos s − sin s
0 sin s cos s
for all s ∈ R.
Linear Lorentzian Rotations around a Spacelike Axis
Suppose that the axis of rotation is the spacelike axis ℓ = {(0, 0, t)
:
t∈
R}. Since Ae3 = e3 and since hAe3 , Aej i = he3 , ej i = 0 for all j ∈ {1, 2},
we get


a1 a2 0



A=
 b1 b2 0 .
0 0 1
Note that −hAe1 , Ae1 i = hAe2 , Ae2 i = 1 and hAe1 , Ae2 i = 0, so
a21 − b21 = 1,
(4.5)
b22 − a22 = 1,
b1 b2 − a1 a2 = 0,
(4.6)
and
a1 b2 − a2 b1 = 1.
(4.7)
(4.8)
Thus a1 = b2 and a2 = b1 . Put ϕ = sinh−1 ◦ b1 , so ϕ ∈ F(R) and
b1 = sinh ϕ. Hence
a21 = 1 + b21 = cosh2 ϕ,
so a1 = cosh ϕ or − cosh ϕ. Because A(s + t) = A(s) · A(t) for all s, t ∈ R,
a1 must be cosh ϕ. Thus


cosh ϕ sinh ϕ 0



A=
 sinh ϕ cosh ϕ 0 .
0
0
1
4.1 Groups of Linear Lorentzian Rotations
53
But sinh is one-to-one, so ϕ is additive. Thus there is an a ∈ R such that
ϕ(s) = as for all s ∈ R. If a 6= 0, the matrix A

cosh s sinh s

A(s) = 
 sinh s cosh s
0
0
for all s ∈ R.
can be given by

0

0

1
Linear Lorentzian Rotations around a Null Axis
Suppose that the axis of rotation is the null axis ℓ = {(t, t, 0)
Since A(e1 + e2 ) = e1 + e2 , it follows that
a2 = 1 − a1 ,
b2 = 1 − b1
:
t ∈ R}.
and c2 = −c1 .
Note that −1 = hAe1 , Ae1 i = −a21 + b21 + c21 and that 1 = hAe2 , Ae2 i =
−a22 + b22 + c22 = −(1 − a1 )2 + (1 − b1 )2 + c21 . We then have b1 = a1 − 1 and
b2 = 2 − a1 . The equation −a21 + b21 + c21 = −1 implies that a1 = 1 + c21 /2.
Now let ϕ = c1 ; we get


A=

1+
ϕ2
2
ϕ2
2
2
− ϕ2
a3
−ϕ
c3
1−
ϕ
ϕ2
2


b3 
.
Because −a23 + b23 + c23 = 1 and hA(e1 + e2 + e3 ), A(e1 + e2 + e3 )i = 1,
we deduce that a3 = b3 and c23 = 1; moreover, c3 is a real constant. But
hAe1 , Ae3 i = 0, so a3 = c3 ϕ. Since det A ≡ 1, it follows that c3 ≡ 1.
Therefore,


A=

1+
ϕ2
2
ϕ
ϕ2
2
2
− ϕ2
1−
ϕ2
2
−ϕ
ϕ


ϕ
.
1
As in the previous cases, there is a real number a such that ϕ(s) = as
for all s ∈ R. If a 6= 0, we can assume that ϕ is the identity function.
4. Lorentzian Motions in E31
54
Consequently,
for all s ∈ R.
4.2


A(s) = 

s2
2
1+
s2
2
s
2
− s2
s
−s
1
1−
s2
2


s

Groups of Lorentzian Motions in E31
By a Lorentzian motion in En1 we mean an isometry in En1 . Similar to the
Euclidean case, the group of Lorentzian motions in En1 is the semidirect
product of the group of translations and the Lorentzian group O1 (n).
Proposition 4.2. ([27], Proposition 9.10). Each Lorentzian motion in
En1 has a unique expression as Tt ◦ A, where Tt is translation by t ∈ En1
and A ∈ O1 (n).
For convenience we will write a vector x ∈ En1 as ( x1 ) ∈ Mn+1,1 (R).
The above proposition implies that the group of Lorentzian motions in
En1 is indeed the Lorentzian affine group
LorAff n (R) =
(Ã
!
A t
0
1
: A ∈ O1(n) and t ∈ En1
)
6 GLn+1 (R).
The matrix subgroup
LorAff n (+, ↑) =
(Ã
!
A t
0
1
: A ∈ Lorn(+, ↑) and t ∈
En1
)
of LorAff n (R) is called the proper orthochronous Lorentzian affine group.
Lemma 4.3. Let γ be any one-parameter group of Lorentzian motions
in En1 . Then γ is a one-parameter group in the proper orthochronous
Lorentzian affine group LorAff n (+, ↑).
4.2 Groups of Lorentzian Motions in E31
55
Proof. Note that for all s ∈ R,
γ(s) =
Ã
!
A(s) t(s)
0
1
∈ LorAff n (R).
Define α : R → GLn (R) by α(s) = A(s) for all s ∈ R. Then α is a smooth
curve in GLn (R). Since γ(0) = In+1 , α(0) = In . But Lorn (+, ↑) is the
connected component of the identity in O1 (n), so
A(s) = α(s) ∈ Lorn (+, ↑)
for all s ∈ R.
Consequently, γ is a one-parameter group in LorAff n (+, ↑).
We could define a Lorentzian screw motion as a Lorentzian rotation
followed by a translation in the direction of the axis of rotation. It turns
out that this definition is suitable whenever the axis of rotation is nonnull.
On the other hand, a Lorentzian rotation around a null axis together with
a translation in the direction of the axis is again a Lorentzian rotation
around a null axis, which is parallel to the former axis—for instance, the
Lorentzian motion

1+



L (s) = 


s2
2
s
2
s2
2
− s2
1−
s2
2
−s
0
0
s hs


s hs 


1 0 

0 1
(where h is a nonzero constant) is exactly the Lorentzian rotation around
the null axis {(x, x, −h)
to
: x ∈ R}; we note further that L (s) is conjugate







1+
s2
2
s
0
s2
2
2
− s2
2
1 − s2
−s
0
s 0


s 0 


1 0 

0 1
4. Lorentzian Motions in E31
56
since

1 0 0
0



 0 1 0 0 


L (s) = 

 0 0 1 −h  


0 0 0 1
1+
s2
2
s
0
s2
2
2
− s2
1−
s2
2
−s
0
s 0

1 0 0 0





s 0 
 0 1 0 0 

.


1 0 
 0 0 1 h 
0 1
0 0 0 1
Even though L : s 7→ L (s) is a one-parameter group of Lorentzian
motions in E31 , many other one-parameter families of translations together
with Lorentzian rotations around a null axis constitute a one-parameter
group of Lorentzian motions. An element of LorAff n (+, ↑) which is not
an exact translation is termed a Lorentzian screw motion.
The following proposition gives a classification of all nontrivial oneparameter groups of Lorentzian motions in E31 .
Proposition 4.4. Up to conjugation in the group of all Lorentzian motions, the following 3 cases cover all possible nontrivial one-parameter
groups of Lorentzian motions in E31 :
   

 
hs
x
1
0
0
x
   

 
y  7→ 0 cos s − sin s y  +  0  ,
   

 
0
z
0 sin s cos s
z
 

   
x
cosh s sinh s 0
x
0
 

   
y  7→  sinh s cosh s 0 y  +  0  ,
 

   
z
0
0
1
z
hs
 

 
 3

2
2
s
x
x
1 + s2
− s2
s
+
s
 

 
 33

s2


y  7→  s2

 s − s .
+
h
1 − 2 s  y 

 
3
 2
2
s
−s
1
s
z
z
(1st type)
(2nd type)
(3rd type)
Proof. Let γ be any nontrivial one-parameter group of Lorentzian motions
in E31 . By Lemma 4.3, γ is a nontrivial one-parameter group in the proper
orthochronous Lorentzian affine group
)
(Ã
!
A t
3
: A ∈ Lor3(+, ↑) and t ∈ E1 6 GL4(R).
LorAff 3 (+, ↑) =
0 1
4.2 Groups of Lorentzian Motions in E31
57
Hence for each s ∈ R, there exist A(s) ∈ Lor3 (+, ↑) and t(s) ∈ E31 such
that
γ(s) ·
à !
x
1
Ã
! Ã ! Ã
!
A(s) t(s)
x
A(s) · x + t(s)
=
·
=
0
1
1
1
for all x ∈ E31 .
Depending on the axis of rotation being timelike, spacelike or null,
the derivative
γ ′ (0) =
is conjugate

0 0

 0 0


 0 1

0 0
Ã
!
A′ (0) t′ (0)
0
0
to
0
−1
0
0
u


0 1 0 u


0
0
1 u






 1 0 0 v 
 0 0 1 v 
v 





 or 
 or 
.





0
0
0
1
−1
0
w 
w
w




0
0 0 0 0
0 0 0 0
By taking a translational conjugation



1 0 0 0
0 0 0 u
1



 0 1 0 w   0 0 −1 v   0






 0 0 1 −v   0 1 0 w   0



0 0 0 1
0 0 0 0
0



0 1 0 u
1
1 0 0 v



 0 1 0 u  1 0 0 v  0






 0 0 1 0  0 0 0 w  0



0 0 0 1
0 0 0 0
0



0 0 1 u
1 0
1 0 0
0



 0 1 0 −w   0 0 1 v   0 1






 0 0 1 u+v   1 −1 0 w   0 0
2



0 0 0
1
0 0 0 0
0 0
0 0
0


1 0 −w 

 or
0 1 v 

0 0 1

0 0 −v

1 0 −u 

 or
0 1 0 

0 0 1

0
0

w 
0

,
u+v 
1 − 2 
0
1
4. Lorentzian Motions in E31
58
γ ′ (0) can be regarded as






0 0 0 h1
0 1 0 0
0 0 1 h3






 0 0 −1 0 
 1 0 0 0 
 0 0 1 −h 
3 






 or 
 or 
,
 0 1 0
 0 0 0 h2 
 1 −1 0
0 
0 






0 0 0
0
0 0 0 0
0 0 0
0
where h1 = u, h2 = w and h3 = (u − v)/2. It follows from Theorem 3.34
that
γ(s) = exp(sγ ′ (0))
for all s ∈ R,
and the proof is complete.
Notes 4.5.
1. In Proposition 4.4, the linear part is a Lorentzian rotation whose
axis of rotation is timelike in the 1st type, spacelike in the 2nd
type, and null in the 3rd type of Lorentzian motions; a Lorentzian
motion of the 3rd type is called a cubic screw motion (or kubische
Schraubung in German).
2. A non cubic screw motion has a property that if we take a point
of the axis of rotation, then the orbit of this point is just the axis
(or the point itself if the screw motion is a rotation). A cubic
screw motion, on the other hand, does not have this property. For
example, the orbit of the origin under the one-parameter group of
cubic screw motions in Proposition 4.4 is a null cubic curve, just
given by the translational parts of the screw motions.
3. The nontrivial one-parameter group of Lorentzian motions of the
1st type has the same assignment
 

   
x
1
0
0
x
hs
 

   
y  7→ 0 cos s − sin s y  +  0 
 

   
z
0 sin s cos s
z
0
4.2 Groups of Lorentzian Motions in E31
59
as the one of a nontrivial one-parameter group of Euclidean motions
in Euclidean 3-space (up to conjugation in the group of Euclidean
motions in E3 ). The Euclidean motions


hs
1
0
0


 0 cos s − sin s 0 




 0 sin s cos s

0


0
0
0
1
are called Euclidean screw motions, where h is a constant.
Chapter
5
Curvatures
In this chapter, we define the Gaussian curvature K, the second Gaussian
curvature KII , the mean curvature H, and the second mean curvature
HII , and determine their formulas. Then, in the next two chapters, we
discuss ruled Weingarten surfaces and ruled linear Weingarten surfaces,
respectively.
Throughout this chapter, we assume that M is a simple surface in E31
determined by a single proper patch x : Ω → E31 with unit surface normal
U=
5.1
xs ×xt
|xs ×xt | .
The Gaussian and Mean Curvatures
The Gaussian Curvature
Since M is of dimension 2, the Gaussian curvature K of M is nothing but
the sectional curvature of M . Note that the ambient space E31 is flat. The
following proposition follows immediately from Corollary 4.20 in [27].
Proposition 5.1. Let S be a shape operator of M derived from U . Then
for all p ∈ M ,
61
62
5. Curvatures
K(p) = hUp , Up i
Therefore,
hS(xs (p)), xs (p)ihS(xt (p)), xt (p)i − hS(xs (p)), xt (p)i2
.
hxs (p), xs (p)ihxt (p), xt (p)i − hxs (p), xt (p)i2
det II
,
(5.1)
det I
where I and II are the first and second fundamental forms of M , respecK = hU, U i
tively.
Remark 5.2. It follows from Weingarten equations (Theorem 1.19) that
K = hU, U i det S .
(5.2)
Brioschi’s Formula
In this subsection we will establish Brioschi’s formula in E31 in order to
show that the Gaussian curvature K of M depends only on the components of the first fundamental form.
Lemma 5.3. The Gaussian curvature K of M is given by
¯
¯
¯
¯

hx
,
x
i
hx
,
x
i
hx
,
x
i

¯ ss tt
ss
s
ss
t ¯

¯
¯
1
¯ hxs , xtt i hxs , xs i hxs , xt i ¯
K=
¯
¯
(EG − F 2 )2 

¯¯ hx , x i hx , x i hx , x i ¯¯
t
tt
t
s
t
t
¯
¯
¯
¯
hx
,
x
i
hx
,
x
i
hx
,
x
i
¯ st st

st
s
st
t ¯
¯
¯
− ¯¯ hxs , xst i hxs , xs i hxs , xt i ¯¯ .
¯

¯
¯ hxt , xst i hxt , xs i hxt , xt i ¯
Proof.
K = hU, U i
det II
det I
®­
® ­
®
­
xs ×xt
xs ×xt 2
xs ×xt
hxs × xt , xs × xt i xss , |xs ×xt | xtt , |xs ×xt | − xst , |xs ×xt |
·
=
|xs × xt |2
EG − F 2
2
EG − F
(xss xs xt )(xtt xs xt ) − (xst xs xt )2
=−
·
|xs × xt |4
EG − F 2
ª
©
−1
· (xss xs xt )(xtt xs xt ) − (xst xs xt )2 .
=
2
2
(EG − F )
5.1 The Gaussian and Mean Curvatures
63
Since
and
¯
¯
¯
¯
hx
,
x
i
hx
,
x
i
hx
,
x
i
¯ ss tt
ss
s
ss
t ¯
¯
¯
(xss xs xt )(xtt xs xt ) = − ¯¯ hxs , xtt i hxs , xs i hxs , xt i ¯¯
¯
¯
¯ hxt , xtt i hxt , xs i hxt , xt i ¯
¯
¯
¯
¯
¯ hxst , xst i hxst , xs i hxst , xt i ¯
¯
¯
(xst xs xt )2 = − ¯¯ hxs , xst i hxs , xs i hxs , xt i ¯¯ ,
¯
¯
¯ hxt , xst i hxt , xs i hxt , xt i ¯
it follows that
¯
¯
¯
¯

hx
,
x
i
hx
,
x
i
hx
,
x
i
¯
¯

ss
tt
ss
s
ss
t
¯
¯
1
¯
K=
hxs , xtt i hxs , xs i hxs , xt i ¯¯
¯
2
2
(EG − F ) 

¯¯ hx , x i hx , x i hx , x i ¯¯
t
tt
t
s
t
t
¯
¯
¯
¯
¯ hxst , xst i hxst , xs i hxst , xt i ¯

¯
¯
¯
¯
− ¯ hxs , xst i hxs , xs i hxs , xt i ¯ .
¯

¯
¯ hx , x i hx , x i hx , x i ¯
t
st
t
s
t
t
We note that hxss , xtt i−hxst , xst i = hxs , xtt is −hxs , xtts i−hxs , xst it +
hxs , xstt i. But xtts = xstt , so
hxss , xtt i − hxst , xst i
=
=
=
=
hxs , xtt is − hxs , xst it
¡
¢
hxs , xt it − hxst , xt i s − 21 hxs , xs itt
hxs , xt its − 21 hxt , xt iss − 12 hxs , xs itt
− 21 Ett + Fst − 12 Gss .
It is easily seen that
hxss , xs i = 21 Es , hxss , xt i = Fs − 12 Et , hxs , xtt i = Ft − 12 Gs ,
hxt , xtt i = 21 Gt , hxst , xs i = 12 Et , and hxst , xt i = 12 Gs .
We then get Brioschi’s formula.
64
5. Curvatures
Theorem 5.4 (Brioschi’s Formula). The Gaussian curvature K of M
is given by
¯
¯ 1


¯ − 2 Ett + Fst − 21 Gss

¯
1
¯
K=
Ft − 21 Gs
¯
(EG − F 2 )2 
¯

¯
1
G
2
¯
¯
¯ 0
¯
− ¯¯ 12 Et
¯1
¯ 2 Gs
t
1
2 Et
E
F
1
2 Es
E
F
¯
¯
1
G
¯
s
2
¯
F ¯¯ .

¯
G ¯
¯
¯
− 12 Et + Fs ¯
¯
¯
F
¯
¯
¯
G
Remark. Brioschi’s formula in Minkowski 3-space is the same as the one
in Euclidean 3-space.
The Mean Curvature
We define the mean curvature vector field H of M as follows: Let p ∈ M
and let {e1 , e2 } be any orthonormal frame on M at p. Define
Hp =
ª
1©
he1 , e1 iII(e1 , e1 ) + he2 , e2 iII(e2 , e2 ) ,
2
where II is the second fundamental form tensor of M . Let HI : M → R
be a smooth real-valued function such that
H = HI U.
(5.3)
Proposition 5.5. The function HI defined above is given by
HI = hU, U i
eG − 2f F + gE
eG − 2f F + gE
=−
2
2(EG − F )
2|EG − F 2 |
(5.4)
Proof. Let p ∈ M be given. Note that, since M is regular, xs (p) and
xt (p) form a basis for the tangent plane Tp M to M at p.
Case 1 : xs is nonnull. Using Gram–Schmidt orthogonalization process, we obtain an orthogonal basis {v1 , v2 } for Tp M , where
v1 = x s
and
v2 = x t −
F
hxs , xt i
xs = xt − xs .
hxs , xs i
E
5.1 The Gaussian and Mean Curvatures
65
We now get an orthonormal basis {e1 , e2 } for Tp M such that
v1
xs
e1 = p
=p
|hv1 , v1 i|
|E|
and
s¯
¯
¯ Ext − F xs
¯
E
¯
¯
= ¯
.
e2 = p
2
EG − F ¯
E
|hv2 , v2 i|
v2
Since hU, U iHI = hH, U i, so
©
ª
1
HI = hU, U i he1 , e1 ihII(e1 , e1 ), U i + he2 , e2 ihII(e2 , e2 ), U i
2
©
ª
1
= hU, U i he1 , e1 ihS(e1 ), e1 i + he2 , e2 ihS(e2 ), e2 i .
2
It is easy to see that
E
,
|E|
e
hS(e1 ), e1 i =
,
|E|
|E| EG − F 2
,
E |EG − F 2 |
gE 2 − 2f EF + eF 2
hS(e2 ), e2 i =
.
|E||EG − F 2 |
he2 , e2 i =
he1 , e1 i =
and
Therefore,
½
¾
e
gE 2 − 2f EF + eF 2
1
+
HI = hU, U i
2
E
E(EG − F 2 )
eG − 2f F + gE
= hU, U i
.
2(EG − F 2 )
Case 2 : xt is nonnull. Similar to Case 1.
Case 3 : xs and xt are null. Since hxs , xt i =
6 0, xs + xt is nonnull.
Applying Gram–Schmidt orthogonalization process to the basis {xs +
xt , xs } for Tp M , we get an orthogonal basis {v1 , v2 } for Tp M such that
v1 = x s + x t
and
v2 = x s −
xs − x t
hv1 , xs i
v1 =
.
hv1 , v1 i
2
We then get an orthonormal basis {e1 , e2 } for Tp M , where
v1
xs + x t
e1 = p
= p
|hv1 , v1 i|
2|F |
and
v2
xs − x t
e2 = p
= p
.
|hv2 , v2 i|
2|F |
66
5. Curvatures
Since
F
,
|F |
e + 2f + g
,
hS(e1 ), e1 i =
2|F |
he1 , e1 i =
F
,
|F |
e − 2f + g
hS(e2 ), e2 i =
,
2|F |
he2 , e2 i = −
and
it follows that
½
¾
1
e + 2f + g e − 2f + g
HI = hU, U i
−
2
2F
2F
1
2f
= hU, U i
2
F
eG − 2f F + gE
= hU, U i
.
(∵ E = G = 0)
2(EG − F 2 )
In either case, we eventually get
HI = hU, U i
eG − 2f F + gE
.
2(EG − F 2 )
By the Lagrange identity in E31 ,
Hence
¯
¯
¯ hx , x i hx , x i ¯
t
s ¯
¯ s s
hxs × xt , xs × xt i = − ¯
¯ = −(EG − F 2 ).
¯ hxs , xt i hxt , xt i ¯
HI := −
eG − 2f F + gE
.
2|EG − F 2 |
Define the mean curvature H of M by H := −HI , so
H=
eG − 2f F + gE
.
2|EG − F 2 |
(5.5)
Remark 5.6. Weingarten equations (Theorem 1.19) imply that
−H = HI = hU, U i
1
tr S .
2
(5.6)
5.2 The Second Gaussian and Second Mean Curvatures
5.2
67
The Second Gaussian and Second Mean Curvatures
We are going to view the second fundamental form II of M as a new
metric tensor on M , so the second Gaussian and second mean curvatures
can be defined formally. Since II is symmetric and bilinear and since M
is connected, it follows that
II is a metric tensor on M if and only if it is nondegenerate.
The following proposition gives a criterion for nondegeneracy of II.
Proposition 5.7. The second fundamental form II of M is nondegenerate if and only if M is nondevelopable (i.e., its Gaussian curvature never
vanishes).
Proof. (⇒) Assume that II is nondegenerate. Let p ∈ M be arbitrary.
Since IIp is a scalar product on Tp M , there exists an orthonormal basis
{u, v} for Tp M with respect to IIp . By Proposition 5.1,
¯
¯
¯
¯ ¯ IIp (u, u)IIp (v, v) − IIp (u, v)2 ¯
1
¯K(p)¯ = ¯
¯= ¯
¯
¯ ¯hu, uihv, vi − hu, vi2 ¯¯ .
hu, uihv, vi − hu, vi2
Hence K(p) 6= 0.
(⇐) Assume that
K = hU, U i
eg − f 2
6= 0.
EG − F 2
Let p ∈ M be given, and let u ∈ Tp M be such that IIp (u, v) = 0 for all
v ∈ Tp M . We have to show that u = 0. Let u = axs + bxt for some real
numbers a and b. We have 2 cases:
Case 1 : f = 0. Thus e 6= 0 and g 6= 0. Since IIp (u, xs ) = 0 and
IIp (xt , xs ) = 0, it follows that 0 = IIp (u, xs ) = ae, so a = 0. Similarly,
b = 0. Therefore, u = 0.
68
5. Curvatures
Case 2 : f 6= 0. Suppose to the contrary that u 6= 0, so a 6= 0. Because
0 = IIp (u, xs ) = ae + bf and 0 = IIp (u, xt ) = af + bg, we conclude that
eg − f 2 = (−b/a)f g − f 2 = (−f /a)(bg + af ) = 0, so K(p) = 0, which is
a contradiction. Therefore, u = 0.
This shows that IIp is nondegenerate.
The second Gaussian Curvature
If M is nondevelopable, we can regard the second fundamental form II
of M as a new metric tensor on M . Likewise, since the semi-Riemannian
manifold (M, II) is of dimension 2, the second Gaussian curvature (or
the inner curvature of the second fundamental form), denoted by KII , is
nothing but the Gaussian curvature of (M, II).
By Theorem 5.4 we are able to compute the second Gaussian curvature KII of M by replacing the components of the first fundamental
form E, F, G by the components of the second fundamental form e, f, g
respectively in Brioschi’s formula.
Proposition 5.8. If M is nondevelopable, then its second Gaussian curvature KII is given by
¯
¯
¯ 1
¯
1
1
1

e
+
f
−
g
e
−
e
+
f
−

¯
¯
tt
st
ss
s
t
s
2
2
2
2

¯
¯
1
1
¯
¯
KII =
ft − 2 gs
e
f
¯
¯
2
2
(eg − f ) 
¯

¯
¯
1
¯
g
f
g
2 t
¯
¯
¯
¯
1
et 12 gs ¯
¯ 0

2
¯
¯
1
¯
¯
− ¯ 2 et e
f ¯ .
¯1

¯
¯ 2 gs f
g ¯
(5.7)
The Second Mean Curvature
b be the Levi-Civita
Suppose that M is nondevelopable. Let ∇ and ∇
connections of the metric tensors I and II, respectively, and let Γkij and
5.2 The Second Gaussian and Second Mean Curvatures
69
b k be the coefficients of ∇ and ∇,
b respectively. The difference tensor T
Γ
ij
is defined by
b k − Γk
Tkij = Γ
ij
ij
for all i, j, k ∈ {1, 2}.
The following proposition follows immediately from Corollary 1.17.
Proposition 5.9. Let i ∈ {1, 2} be given. Then:
¢
¡ p
b k = ln |eg − f 2 | .
(i) Γ
ik
|i
¡ p ¢
(ii) Tkik = ln |K| |i , where K is the Gaussian curvature of M .
For convenience we denote the second fundamental form II by
II = Lij dxi dxj .
(5.8)
Since II is nondegenerate, the matrix (Lij ) is invertible. Let (Lij ) denote the inverse matrix of (Lij ). This leads to the relation between the
Gaussian curvature and the mean curvature of M .
Lemma 5.10. KLij gij = 2HI = −2H.
Proof.
µ
¶
gE − 2f F + eG
eg − f 2
EG − F 2
eg − f 2
gE − 2f F + eG
= 2hU, U i
2(EG − F 2 )
KLij gij = hU, U i
= 2HI = −2H.
Recall that M is a nondevelopable simple surface determined by the
proper patch x : Ω → E31 with unit surface normal U =
xs ×xt
|xs ×xt | .
Let D
be a bounded connected open set whose closure D is contained in Ω, let
γ : D → R be a C2 -function such that γ ≡ γs ≡ γt ≡ 0 on the boundary
¯
f := x(D) be a portion of M determined by x¯ : D → E3 .
of D and let M
D
1
70
5. Curvatures
f
Suppose that ε ∈ R+ . The normal variation ϕ : D× ] − ε, ε [ → E31 of M
determined by γ is given by
ϕ(s, t, v) = x(s, t) + vγ(s, t)U (s, t)
for all (s, t) ∈ D and all v ∈ ] − ε, ε [ .
For each v ∈ ] − ε, ε [ , define xv : D → E31 by
xv (s, t) = ϕ(s, t, v) for all (s, t) ∈ D.
If ε is small enough, we can assume that for every v ∈ ] − ε, ε [ , M v :=
xv (D) is a portion of a nondevelopable surface determined by xv with
(i)
∂xv
= xs + vγUs + vγs U ,
∂s
(ii)
∂xv
= xt + vγUt + vγt U ,
∂t
and
(iii) the II-area of M v which is denoted by AvII and defined as the integral
ZZ q
¯
¯
v
¯det II v ¯ dsdt,
AII =
D
where II v is the second fundamental form of M v .
Let v ∈ ] − ε, ε [ be fixed. If we denote the components of the first
fundamental form of M v by E v , F v , Gv , then we obtain that
¡
¢
E v = E + vγ hxs , Us i + hxs , Us i + v 2 γ 2 hUs , Us i + hU, U iv 2 γs2 ,
¡
¢
F v = F + vγ hxs , Ut i + hxt , Us i + v 2 γ 2 hUs , Ut i + hU, U iv 2 γs γt ,
¡
¢
Gv = G + vγ hxt , Ut i + hxt , Ut i + v 2 γ 2 hUt , Ut i + hU, U iv 2 γt2 .
Hence
¡ ¢2
E v Gv − F v = EG − F 2 − 2vγ(gE − 2f F + eG) + O(v 2 )
= EG − F 2 − 4vγ
gE − 2f F + eG
(EG − F 2 ) + O(v 2 )
2(EG − F 2 )
= (EG − F 2 )(1 − 4hU, U ivγHI ) + O(v 2 ).
5.2 The Second Gaussian and Second Mean Curvatures
71
Here the notation O(η) means for all ε > 0, there exists an R ∈ R+
such that
|O(η)| ≤ R|η|
for all η ∈ R with |η| < ε.
Note that
¡
¢
sgn E v Gv − (F v )2 = sgn(EG − F 2 ),
where the signum function sgn is defined by



 1 if x > 0,
sgn(x) =
0 if x = 0,


 −1 if x < 0.
­
®
It follows that U v , U v = hU, U i and that
¯
¯
¡
¢
¯ v v ¡ v ¢2 ¯
¯E G − F ¯ = −hU, U i (EG − F 2 )(1 − 4hU, U ivγHI ) + O(v 2 ) .
Note that
xvs × xvt
Uv = q
¯
¡ ¢ ¯
¯E v Gv − F v 2 ¯
(
xs × xt + vγ(xs × Ut + Us × xt )
+vγs (U × xt ) + vγt (xs × U ) + O(v 2 )
and that
)
= q¯
¯
¯(EG − F 2 )(1 − 4hU, U ivγHI ) + O(v 2 )¯
xvss = xss + vγUss + 2vγs Us + vγss U,
xvst = xst + vγUst + vγt Us + vγs Ut + vγst U,
xvtt = xtt + vγUtt + 2vγt Ut + vγtt U.
72
5. Curvatures
Hence
¿
À
∂ 2 xv v
e =
,U
∂s2


¡
¢


hx
,
x
×
x
i
+
vγ
hx
,
x
×
U
i
+
hx
,
U
×
x
i


ss
s
t
ss
s
t
ss
s
t


+vγs hxss , U × xt i + vγt hxss , xs × U i + vγhUss , xs × xt i





 +vγ hU, x × x i + O(v 2 )
ss
s
t
=
,
q¯
¯
¯(EG − F 2 )(1 − 4hU, U ivγHI ) + O(v 2 )¯
À
¿ 2 v
∂ x
v
v
,U
f =
∂s∂t


¡
¢


hx
,
x
×
x
i
+
vγ
hx
,
x
×
U
i
+
hx
,
U
×
x
i


t
st
s
t
st
s
t

 st s
v
and
+vγs hxst , U × xt i + vγt hxst , xs × U i + vγhUst , xs × xt i





 +vγ hU, x × x i + O(v 2 )
st
s
t
=
q¯
¯
¯(EG − F 2 )(1 − 4hU, U ivγHI ) + O(v 2 )¯
¿
À
∂ 2 xv v
g =
,U
∂t2

¡
¢


 hxtt , xs × xt i + vγ hxtt , xs × Ut i + hxtt , Us × xt i
+vγs hxtt , U × xt i + vγt hxtt , xs × U i + vγhUtt , xs × xt i


 +vγ hU, x × x i + O(v 2 )
tt
s
t
=
q¯
¯
¯(EG − F 2 )(1 − 4hU, U ivγHI ) + O(v 2 )¯
v
¯
∂ ¯¯
To investigate
ev :
¯
∂v
v=0
¯
∂ ¯¯
v
e
=
∂v ¯







.
v=0













p
¡
¢
|EG − F 2 | (xss xs Ut ) + (xss Us xt ) + (Uss xs xt )
p
p
¡
¢
+ |EG − F 2 | γs (xss U xt ) + γt (xss xs U ) + hU, U iγss |EG − F 2 |
γ
+2hU, U iγeHI |EG −
F 2|
¯
¯
¯EG − F 2 ¯













.
5.2 The Second Gaussian and Second Mean Curvatures
73
By Theorem 1.19 (Weingarten equations), we obtain that
(i)
F f − Eg
hxss , xs × xt i
EG − F 2
(Eg − F f )e
= hU, U i q¯
¯,
¯EG − F 2 ¯
hxss , xs × Ut i =
(ii)
F f − Ge
hxss , xs × xt i
EG − F 2
(Ge − F f )e
= hU, U i q¯
¯,
¯EG − F 2 ¯
hxss , Us × xt i =
(iii)
q¯
¯
¯EG − F 2 ¯hUss , U i
q¯
¯
= − ¯EG − F 2 ¯hUs , Us i
q¯
¯
¯EG − F 2 ¯
=−
(EG − F 2 )(Ef 2 + Ge2 − 2F ef )
(EG − F 2 )2
hU, U i
2
2
= q¯
¯ (Ef + Ge − 2F ef ).
¯EG − F 2 ¯
hUss , xs × xt i =
Therefore,
q¯
¯¡
¢
γ ¯EG − F 2 ¯ (xss xs Ut ) + (xss Us xt ) + (Uss xs xt )
¯
¯
+ 2hU, U iγeHI ¯EG − F 2 ¯
¯
¯¢
¡
= hU, U iγ 2e(Eg − 2F f + Ge) − E(eg − f 2 ) + 2eHI ¯EG − F 2 ¯
¯
¯¢
¡
= hU, U iγ 4ehU, U iHI (EG−F 2 )−hU, U iEK(EG−F 2 )+2eHI ¯EG−F 2 ¯
¯
¯¢
¯
¯
¯
¯
¡
= hU, U iγ −4eHI ¯EG − F 2 ¯ + EK ¯EG − F 2 ¯ + 2eHI ¯EG − F 2 ¯
¯
¡
¢¯
= hU, U iγ EK − 2eHI ¯EG − F 2 ¯.
74
5. Curvatures
From Theorem 1.18 (Gauss equations), we know that
xss = Γ111 xs + Γ211 xt + hU, U ieU.
Thus
¡
¯
¯
¯
¢q¯
γs hxss , U × xt i + γt hxss , xs × U i ¯EG − F 2 ¯ + hU, U iγss ¯EG − F 2 ¯
¯
¯
¯
¡
¢q¯
= −γs hxss × xt , U i − γt hxs × xss , U i ¯EG − F 2 ¯ + hU, U iγss ¯EG − F 2 ¯
¯
¡
¢q¯
= −γs Γ111 hxs × xt , U i − γt Γ211 hxs × xt , U i ¯EG − F 2 ¯
¯
¯
+hU, U iγss ¯EG − F 2 ¯
¯
¢¯
¡
= hU, U i −γs Γ111 − γt Γ211 + γss ¯EG − F 2 ¯
¯
¡
¢¯
= hU, U i ∇1 γ|1 ¯EG − F 2 ¯.
We deduce that
¯
∂ ¯¯
ev = hU, U i(γKE + ∇1 γ|1 − 2γHI e).
∂v ¯v=0
In other words,
¯
∂ ¯¯
∂v ¯
v=0
Lv11 = hU, U i(γKg11 + ∇1 γ|1 − 2γHI L11 ).
(5.9)
Similarly,
and
¯
∂ ¯¯
Lv = hU, U i(γKg12 + ∇1 γ|2 − 2γHI L12 )
∂v ¯v=0 12
¯
∂ ¯¯
Lv = hU, U i(γKg22 + ∇2 γ|2 − 2γHI L22 ).
∂v ¯v=0 22
In conclusion,
¯
∂ ¯¯
Lv = hU, U i(γKgij + ∇i γ|j − 2γHI Lij )
∂v ¯v=0 ij
(5.10)
(5.11)
5.2 The Second Gaussian and Second Mean Curvatures
75
for all i, j ∈ {1, 2}. Note that for each v ∈ ] − ε, ε [ , if ζ v = sgn(det II v ),
then
¢
∂ ¡p
| det II v |
∂v
¢
∂ ¡
1
1
| det II v |
= p
2 | det II v | ∂v
¡
¢2 ´
ζv
1
∂ ³ v v
L11 L22 − Lv12
= p
2 | det II v | ∂v
´
³
ζv
∂
∂
1
∂
= p
Lv22 Lv11 − 2Lv12 Lv12 + Lv11 Lv22
2 | det II v |
∂v
∂v
∂v
³
´
v
v
1 ζ det II
∂
∂
∂
= p
(Lv )11 Lv11 + 2(Lv )12 Lv12 + (Lv )22 Lv22
2 | det II v |
∂v
∂v
∂v
p
∂
1
| det II v | (Lv )ij Lvij .
=
2
∂v
This implies that
¯
∂ ¯¯ p
| det II v |
∂v ¯v=0
¯
¯
1p
ij ∂ ¯
| det II| L
=
Lv
2
∂v ¯v=0 ij
¢p
¡
1
| det II|
= hU, U i γKLij gij + Lij ∇i γ|j − 2γHI Lij Lij
2
µ
¶
p
1
| det II|. (by Lemma 5.10)
= hU, U i −γHI + Lij ∇i γ|j
2
In order to define the second mean curvature HII of M , we need the
following 2 lemmas:
Lemma 5.11.
ZZ
D
ij
L ∇i γ|j dAII = −
ZZ
D
¡
¢
b k Lij Tkij dAII .
γ∇
c denote the Laplacian and the divergence with reb and div
Proof. Let ∆
spect to II, respectively. Note that for all i, j ∈ {1, 2},
b i γ|j
∇i γ|j − ∇
b kij γ|k ) = Tkij γ|k .
= (γ|i|j − Γkij γ|k ) − (γ|i|j − Γ
76
5. Curvatures
Thus
ZZ
D
ij
L ∇i γ|j dAII =
=
=
ZZ
Z ZD
Z ZD
D
=
ZZ
D
¢
¡
b i γ|j + Tkij γ|k dAII
Lij ∇
¡
¢
b γ + Lij Tkij γ|k dAII
∆
Lij Tkij γ|k dAII
(by the divergence theorem)
¡
¢
dγ Lij Tkij ∂k dAII .
By the divergence theorem again, we get
ZZ
¡
¢
c γLij Tk ∂k dAII
0=
div
ij
Z ZD
ZZ
¡ ij k ¢
=
dγ L Tij ∂k dAII +
D
D
¢
¡
c Lij Tk ∂k dAII ,
γ div
ij
(by Proposition 1.23)
so
ZZ
D
ij
L ∇i γ|j dAII = −
=−
ZZ
Z ZD
D
¡
¢
c Lij Tk ∂k dAII
γ div
ij
¡
¢
b k Lij Tk dAII .
γ∇
ij
Lemma 5.12.
1
(i) Tkij = Llk ∇j Lil
2
(by (2.10))
for all i, j, k ∈ {1, 2}.
b k Tk = Lij ∇
b j Tk .
(ii) Lij ∇
ij
ik
Proof. (i) We first claim that for all i, j, l ∈ {1, 2},
1
Tsij Lsl = ∇j Lil .
2
Let i, j, l ∈ {1, 2} be arbitrary. From (2.11), we get
∇j Lil = ∂j (Lil ) − II(∇j ∂i , ∂l ) − II(∂i , ∇j ∂l ).
5.2 The Second Gaussian and Second Mean Curvatures
77
Thus
∇j Lil = Lil|j − Γsij Lsl − Γslj Lsi .
(5.12)
The Ricci lemma implies that
b j Lil = Lil|j − Γ
b sij Lsl − Γ
b s Lsi .
0=∇
lj
(5.13)
∇j Lil = Tsij Lsl + Tslj Lsi .
(5.14)
Subtracting (5.13) from (5.12), we get
It follows from (5.14) and the Codazzi equation that
Tsij Lsl + Tsil Lsj = ∇i Ljl = ∇l Lij = Tsli Lsj + Tslj Lsi ,
so
Tsij Lsl = Tslj Lsi .
The equation (5.14) becomes
1
Tsij Lsl = ∇j Lil .
2
This proves the claim.
Therefore, for all i, j, k ∈ {1, 2},
Tkij
= Tsij Lsl Llk
=
1 lk
L ∇j Lil .
2
(ii)
¢
¡
b k Llk ∇j Lil
b k Tkij = 1 Lij ∇
Lij ∇
2
¢
¡
1
b k Lil ∇j Llk
= − Lij ∇
2
¢
1 ij b ¡
= − L Lil ∇k ∇j Llk
2
¢
1b ¡
= − ∇k ∇j Ljk .
2
Similarly,
(∵ ∇j (Lil Llk ) = ∇j δ ki = 0)
¢
¡
b j Tk = − 1 ∇
b j ∇k Ljk .
Lij ∇
ik
2
(by the Ricci lemma)
78
5. Curvatures
We infer that
b j Tk .
b k Tkij = Lij ∇
Lij ∇
ik
This completes the proof.
Hence the first variation of AII is
¯
¯
¾
½Z Z
p
∂ ¯¯
∂ ¯¯
v
v
| det II | dsdt
A =
∂v ¯v=0 II
∂v ¯v=0
D
¾
ZZ ½ ¯
p
∂ ¯¯
v
=
| det II | dsdt
∂v ¯v=0
D
¾
½
ZZ
1 ij
= hU, U i
−γHI + L ∇i γ|j dAII
2
D
¾
ZZ ½
1 b ³ ij k ´
γH − γ ∇k L Tij
= hU, U i
dAII
2
D
(by Lemma 5.11)
½
¾
ZZ
1
b k Tk dAII .
γ H − Lij ∇
= hU, U i
ij
2
D
(by the Ricci lemma)
Define
Since
1
b k Tkij .
HII := H − Lij ∇
2
b k Tkij = Lij ∇
b j Tk
Lij ∇
ik
¢
¡ p
ij b
= L ∇j ln |K| |i
¢
¡ p
b ln |K| ,
=∆
(5.15)
(by Lemma 5.12 (ii))
(by Proposition 5.9 (ii))
(by (2.10))
it follows from (5.15) that
¢
1b¡ p
HII = H − ∆
ln |K| .
2
We now have a formula for HII as follows:
¶
X ∂ µp
¢
¡ p
1
ij ∂
| det II| L
, (5.16)
HII = H− p
ln |K|
∂uj
2 | det II| ij ∂ui
where u1 and u2 stand for “s” and “t,” respectively.
5.3 Curvatures of Ruled Surfaces in E31
79
Definition 5.13. A nondevelopable surface is said to be II-flat [resp.,
II-minimal ] if KII ≡ 0 [resp., HII ≡ 0].
Curvatures of Ruled Surfaces in E31
5.3
Here, we determine formulas of the curvatures K, KII , H and HII of a
nondevelopable ruled surface M in E31 . We have 3 cases to consider according to the type of M itself:
Case 1 : M ∈ M11 . Then we can choose a parametrization x(s, t) =
α(s) + tβ(s) of the ruled surface M such that
hβ, βi = ε, hβ ′ , β ′ i = η, hα′ , β ′ i = 0, and ε, η ∈ {−1, 1}.
Hence α is the striction line of M . The components of the first fundamental form are
E = hα′ , α′ i + ηt2 , F = hα′ , βi, G = ε.
In the moving frame β, β ′ , β × β ′ with signs ε, η, −εη, we get
α′ = εF β − εηQβ × β ′ ,
(5.17)
where the invariant Q := (α′ ββ ′ ) is the parameter of distribution. Note
that EG − F 2 = −ηQ2 + εηt2 , and that
β ′′ = εη(−β + Jβ × β ′ ),
(5.18)
where the invariant J := (β ′′ β ′ β) is the geodesic curvature function, and
the unit normal vector field is
1
(ηQβ ′ − tβ × β ′ ).
U=p
2
2
|Q − εt |
Hence the components of the second fundamental form are
e= p
1
Q
, g = 0.
(εQ(F − QJ) − Q′ t + Jt2 ), f = p
|Q2 − εt2 |
|Q2 − εt2 |
80
5. Curvatures
Putting λ := F/Q, we obtain that
Q2
,
(Q2 − εt2 )2
J 4
t + ε(λ − 2J) t2 + 2εQ′ t + Q2 (λ + J)
1 Q2
,
KII =
¯
¯
2
¯Q2 − εt2 ¯3/2
K=
H=
HII =
1 εJt2 − εQ′ t − Q2 (λ + J)
,
¯
¯
2
¯Q2 − εt2 ¯3/2
1
2
−
and
2J 4
t + ε(5J + 2λ) t2 + 3εQ′ t + Q2 (λ − 3J)
Q2
.
¯
¯
¯Q2 − εt2 ¯3/2
(5.19)
(5.20)
(5.21)
(5.22)
Note. From (5.17) and (5.18), the quantities Q, J, F determine α and β
completely.
The following lemma gives a necessary condition when Q, J, F become
constant.
Lemma 5.14. ([10], pp. 315–316). Let M be a ruled surface of class M11
with parametrization mentioned above. If the invariants Q, J, F are all
constant, then M is a helicoidal ruled surface.
For convenience, we give a detailed proof of this lemma in Appendix
B.
Case 2 : M ∈ M01 . We may assume without loss of generality that M
can be parametrized by x(s, t) = α(s)+tβ(s), where hβ, βi = 1, hβ ′ , β ′ i =
0, and F = hα′ , βi = 0. Since β × β ′ and β ′ are orthogonal null vectors,
they are collinear. We can assume further that β × β ′ = β ′ . Note that β ′
is a null direction in the hyperboloid of one sheet {ν | hν, νi = 1}, so β is
a straight line. Changing the parameter s (if necessary), we get β ′′ = 0.
Put Q := (α′ ββ ′ ); then Q = hα′ , β ′ i. We will see later that Q 6= 0.
Now let R := hα′ , α′ i. By passing to another base curve α + t̃β for some
real number t̃ 6= 0, we can assume that R 6= 0. In the moving frame
5.3 Curvatures of Ruled Surfaces in E31
√1 α′ , β, √1 α′ × β with signs
|R|
|R|
β′ =
R
R
|R| , 1, − |R| ,
81
we obtain that
Q ′
(α − α′ × β).
R
It follows that
(i) Q′ = hα′′ , β ′ i =
Q ′′ ′
Q
hα , α i − (α′′ α′ β) ,
R
R
R′ RQ′
−
,
and
2
Q
³ Q′
R′ ′
R′ ´ ′
(iii) α′′ =
α − Qβ +
−
α ×β.
2R
Q
2R
(ii) (α′′ α′ β) =
The components of the first fundamental form are E = R + 2Qt, F =
0, G = 1. Thus EG − F 2 = R + 2Qt. The unit normal vector field is
n³
Q ´
Q o
1
1 + t α′ × β − tα′ .
U=p
R
R
|R + 2Qt|
Hence the components of the second fundamental form are
e= p
1
|R + 2Qt|
³ R′
2
−
We then have
´
RQ′
Q
− Q′ t , f = q¯
¯ , and g = 0.
Q
¯R + 2Qt¯
(5.23)
Q2
,
(R + 2Qt)2
RQ′ − R′ Q
KII =
,
2Q|R + 2Qt|3/2
−2RQ′ + R′ Q − 2QQ′ t
H=
,
4Q|R + 2Qt|3/2
2RQ′ − 3R′ Q − 2QQ′ t
HII =
.
4Q|R + 2Qt|3/2
K=
(5.24)
(5.25)
and
(5.26)
(5.27)
Note 5.15. In this case, we can apply a Lorentzian motion so that β =
(s, s, 1). Then the base curve α(s) = (α1 (s), α2 (s), α3 (s)) can be reconstructed from the following equations:
82
5. Curvatures
(α2′ − α1′ )(α2′ + α1′ ) + (α3′ )2 = hα′ , α′ i = R,
(α2′ − α1′ )s + α3′ = hα′ , βi = 0,
α2′ − α1′ = hα′ , β ′ i = Q.
It follows that
α′ =
´
Q³ R
2 R
2
−
1
−
s
,
+
1
−
s
,
−2s
.
2 Q2
Q2
(5.28)
This implies that the invariants Q and R completely determine the ruled
surface in question.
Convention. If the ruled surfaces in question are of class M01 , we assume
that they have the same parametrizations as the one in the above discussion with director vector field β = (s, s, 1). As a result, we can construct
these surfaces whenever we are given the invariants Q and R.
Case 3 : M ∈ M0 . We can find a parametrization x(s, t) = α(s) +
tβ(s) of M such that hα′ , β ′ i = 0, hβ, βi = 0, hβ ′ , β ′ i = 1 and β ′ × β = β.
But det I = −F 2 , so F 6= 0. Hence U = {α′ × β + βt}/|F | and the
components of the second fundamental form are
ª
¡
¢
1 © ′′ ′
(α α β) + hα′′ , βi + (α′ ββ ′′ ) t − t2 ,
|F |
1
F
f=
hβ ′ , α′ × β + βti = −
, and g = 0.
|F |
|F |
e=
It follows that
K=
1
1
1
1
, KII = −
, H=
, and HII =
.
2
F
|F |
|F |
|F |
Remarks 5.16.
1. On a nondevelopable ruled surface, the invariant Q := (α′ ββ ′ ) never
vanishes.
2. If M ∈ M11 or M ∈ M01 , then:
(2.1) H = 0 ⇐⇒ HII = 0.
5.3 Curvatures of Ruled Surfaces in E31
83
(2.2) If H = 0 or HII = 0, then KII = 0. The converse of this
statement is true if M ∈ M11 , but it is not true when M ∈ M01 ,
e.g., a Nomizu–Sasaki surface
¡
exp(−s) + t exp(s), − exp(−s) + t exp(s), 2 ln 2 + t − 2s
¢
is II-flat, but it is neither minimal nor II-minimal. (See also
Example 6.7.)
3. If M ∈ M0 , then all curvatures K, KII , H and HII are functions of
the parameter s only, and
K
2
= KII
2
= H 2 = HII
.
Chapter
6
Ruled Weingarten Surfaces
The main purpose of this chapter is to classify nondevelopable ruled surfaces in E31 on which there is a nontrivial functional relation between
a pair of curvatures K, KII , H and HII . For a pair {A, B}, A 6= B, of
the curvatures K, KII , H and HII of a nondevelopable surface M , M is
said to be an {A, B}-W-surface if there is a nontrivial functional relation
Φ(A, B) = 0. A nondevelopable surface is called a Weingarten surface
(or briefly, a W-surface) if it is an {A, B}-W-surface for some pair {A, B}
of K, KII , H and HII .
6.1
Helicoidal Ruled Surfaces
Before getting into the territory of nondevelopable ruled surfaces of Weingarten type, we need to know some examples of these surfaces and relations between their curvatures as models for our classifications.
Definition 6.1. A helicoidal surface M in E3 [resp., E31 ] is the orbit of
a planar curve β = β(t) under a one-parameter group of Euclidean screw
motions [resp., of Lorentzian screw motions]. If the planar curve β is a
straight line, then M is termed a helicoidal ruled surface.
85
86
6. Ruled Weingarten Surfaces
Remark 6.2. It is obvious that any helicoidal surface (not necessarily
ruled) is a Weingarten surface because all curvatures K, KII , H and HII
depend only on the parameter t. This includes the case of surfaces of
revolution when the one-parameter group is of pure rotations.
It is well known that nondevelopable minimal ruled surfaces in E31 are
Lorentzian helicoids (i.e., helicoids of the 1st kind, of the 2nd kind or of
the 3rd kind) and a conjugate of Enneper’s surface of the 2nd kind. All
of these surfaces are helicoidal ruled surfaces.
Example 6.3 (Helicoid of the 1st kind). For a constant a 6= 0, let
M be a ruled surface in E31 with parametrization
x(s, t) = (as, t cos s, t sin s).
Then M is a noncylindrical ruled surface of type M11 , called a helicoid of
the 1st kind, whose curvatures are
K=
a2
(a2 − t2 )2
and KII = H = HII = 0.
Example 6.4 (Helicoid of the 2nd kind). For a constant a 6= 0, let
M be a ruled surface in E31 with parametrization
x(s, t) = (t sinh s, t cosh s, as).
Then M is a noncylindrical ruled surface of type M11 , called a helicoid of
the 2nd kind, whose curvatures are
K=
(a2
a2
− t2 )2
and KII = H = HII = 0.
Note that the above helicoid of the 2nd kind satisfies the equation
x1 = x2 tanh(x3 /a) .
Example 6.5 (Helicoid of the 3rd kind). For a constant a 6= 0, let
M be a ruled surface in E31 with parametrization
x(s, t) = (t cosh s, t sinh s, as).
6.1 Helicoidal Ruled Surfaces
87
Then M is a noncylindrical ruled surface of type M11 , called a helicoid of
the 3rd kind, whose curvatures are
K=
a2
(a2 + t2 )2
and KII = H = HII = 0.
Note that the above helicoid of the 3rd kind satisfies the equation
x2 = x1 tanh(x3 /a) .
Example 6.6 (Conjugate of Enneper’s surface of the 2nd kind).
For a constant a 6= 0, let M be a ruled surface in E31 with parametrization
´
³ s3
´
´
³ ³ s3
+ s + ts, a
− s + ts, as2 + t .
x(s, t) = a
3
3
Then M is a noncylindrical ruled surface of type M01 , called a conjugate
of Enneper’s surface of the 2nd kind or Cayley–Lie minimal ruled surface
of third degree, whose curvatures are
K=
1
4t2
and KII = H = HII = 0.
Note that the above Conjugate of Enneper’s surface of the 2nd kind
satisfies the equation
¢
¡
6a2 (x1 + x2 ) = (x1 − x2 ) 6ax3 − (x1 − x2 )2 .
There is a two-parameter family of helicoidal ruled surfaces which are
called Nomizu–Sasaki surfaces.
Example 6.7 (Nomizu–Sasaki). Let A and B be constants with A 6= 0.
Let M be a ruled surface in E31 which is the orbit of the line
¾
½³
´
A
A
+ t, − + t, B ln |A| + t : t ∈ R
2
2
under the Lorentzian screw motions of the 2nd type with h = −B. Then
M is a noncylindrical ruled surface of type M01 , called a Nomizu–Sasaki
surface, with parametrization
³A
´
A
x(s, t) =
exp(−s)+t exp(s), − exp(−s)+t exp(s), B ln |A|+t−Bs .
2
2
88
6. Ruled Weingarten Surfaces
The curvatures of this surface are
K=
1
,
(2t − A + 2B)2
sgn(A) (2t − 2A + 3B)
H= p
,
2 |A| |2t − A + 2B|3/2
sgn(A) (A − B)
,
KII = p
|A| |2t − A + 2B|3/2
sgn(A) (2t + 2A − B)
HII = p
.
2 |A| |2t − A + 2B|3/2
An easy computation shows that M satisfies the equation
x3 =
1 2
(x − x22 ) + B ln |x1 − x2 | .
2A 1
This surface is a hyperbolic paraboloid if B = 0, in which case it is the
orbit of a straight line under a Lorentzian rotation around the x3 -axis.
In Euclidean 3-space, a (right) conoid is a ruled surface with rulings
parallel to a plane and passing through a straight line that is perpendicular to the plane. In general, conoids are not helicoidal ruled surfaces.
Likewise, up to Lorentzian motions, there are 3 categories of nondevelopable Lorentzian conoids in E31 as follows:
Example 6.8 (Conoid of the 1st kind). Let φ = φ(s) be a smooth
function such that φ′ never vanishes and let M be a ruled surface in E31
with parametrization
¡
¢
x(s, t) = φ(s), t cos s, t sin s .
Then M is a noncylindrical ruled surface of type M11 , called a conoid of
the 1st kind, whose curvatures are
(φ′ (s))2
,
((φ′ (s))2 − t2 )2
φ′′ (s) t
1
,
H=−
2 |(φ′ (s))2 − t2 |3/2
K=
φ′′ (s) t
,
|(φ′ (s))2 − t2 |3/2
φ′′ (s) t
3
HII =
.
2 |(φ′ (s))2 − t2 |3/2
KII =
Example 6.9 (Conoid of the 2nd kind). Let φ = φ(s) be a smooth
function such that φ′ never vanishes and let M be a ruled surface in E31
with parametrization
¡
¢
x(s, t) = t sinh s, t cosh s, φ(s) .
6.1 Helicoidal Ruled Surfaces
89
Then M is a noncylindrical ruled surface of type M11 , called a conoid of
the 2nd kind, whose curvatures are
(φ′ (s))2
,
((φ′ (s))2 − t2 )2
φ′′ (s) t
1
H=
,
2 |(φ′ (s))2 − t2 |3/2
K=
φ′′ (s) t
,
|(φ′ (s))2 − t2 |3/2
φ′′ (s) t
3
HII = −
.
2 |(φ′ (s))2 − t2 |3/2
KII = −
Example 6.10 (Conoid of the 3rd kind). Let φ = φ(s) be a smooth
function such that φ′ never vanishes and let M be a ruled surface in E31
with parametrization
¡
¢
x(s, t) = t cosh s, t sinh s, φ(s) .
Then M is a noncylindrical ruled surface of type M11 , called a conoid of
the 3rd kind, whose curvatures are
K=¡
H=
Notes 6.11.
(φ′ (s))2
(φ′ (s))2 + t2
¢2 ,
φ′′ (s) t
1
,
¡
2 (φ′ (s))2 + t2 ¢3/2
KII = − ¡
φ′′ (s) t
¢3/2 ,
(φ′ (s))2 + t2
φ′′ (s) t
3
HII = − ¡
.
2 (φ′ (s))2 + t2 ¢3/2
1. The helicoid of the 3rd kind and the conoid of the 3rd kind are
timelike, but the other helicoids and conoids as well as a conjugate
of Enneper’s surface of the 2nd kind have a spacelike and a timelike
piece.
2. The curvatures KII , H, HII of the Nomizu–Sasaki surface satisfy
HII − H = 2KII .
3. The curvatures KII , H, HII of each conoid in Examples 6.8–6.10
satisfy 3KII = −6H = 2HII and 2KII + H − HII = 0.
90
6. Ruled Weingarten Surfaces
6.2
Ruled Weingarten Surfaces in E3
In this section we summarize results of W. Kühnel [20] and G. Stamou
[31], concerning classifications of nondevelopable ruled Weingarten surfaces in Euclidean 3-space:
Theorem 6.12 (Classification of ruled {A, B}-W-surfaces in E3 ,
where (A, B) ∈ {(KII , H), (KII , HII ), (H, HII )}
)}).
Let (A, B) ∈ {(KII , H), (KII , HII ), (H, HII )}, and let M : x(s, t) = α(s)+
tβ(s) be a nondevelopable ruled surface in E3 . Then M is a ruled {A, B}-
W-surface if and only if it is one of the following surfaces:
1. a helicoidal ruled surface;
2. a conoid;
3. a surface with invariants
Q(s) =
c1
,
s
J(s) =
c2
,
s
F (s) =
c3
,
s2
where c1 , c2 , c3 are constants with c1 6= 0 and c22 + c23 6= 0, and s is
an arc-length parameter of the regular director curve β.
Moreover, if M is minimal, then it is a helicoid.
Note. The function λ defined in [31] has been simplified by setting λ =
F/Q here.
Theorem 6.13 (Classification of ruled {A, B}-W-surfaces in E3 ,
where (A, B) ∈ {(K, KII ), (K, H), (K, HII )}
)}).
Let (A, B) ∈ {(K, KII ), (K, H), (K, HII )}, and let M : x(s, t) = α(s) +
tβ(s) be a nondevelopable ruled surface in E3 . Then M is a ruled {A, B}-
W-surface if and only if it is a helicoidal ruled surface. Moreover, if M
is minimal, then it is a helicoid.
6.3 Ruled Weingarten Surfaces in E31
6.3
91
Ruled Weingarten Surfaces in E31
We now give classifications of nondevelopable ruled Weingarten surfaces
in Minkowski 3-space. It turns out that the classifications in Minkowski
case are similar to the ones in Euclidean case.
Theorem 6.14 (Classification of ruled {H, HII }-W-surfaces in
E31 ). let M : x(s, t) = α(s) + tβ(s) be a nondevelopable ruled surface in
E31 . Then M is a ruled {H, HII }-W-surface if and only if it satisfies one
of the following properties:
1. M ∈ M11 , and it is one of the following surfaces:
(1.1) a helicoidal ruled surface;
(1.2) a Lorentzian conoid;
and
(1.3) a surface with invariants
Q(s) =
c1
,
s
J(s) =
c2
,
s
F (s) =
c3
,
s2
where c1 , c2 , c3 are constants with c1 6= 0 and c22 + c23 6= 0, and
s is an arc-length parameter of the regular director curve β.
Moreover, if M is minimal, then it is a Lorentzian helicoid.
2. M ∈ M01 is a surface with nonvanishing invariants Q(s), R(s) such
that either Q is constant and R is arbitrary, or Q is arbitrary and
R is given by
R = aQ + bQ
for some constants a, b .
Z
(Q′ )3
ds
Q4
3. M ∈ M0 .
Proof. Since all nondevelopable ruled surfaces in E31 with null rulings are
ruled {H, HII }-W-surfaces, it remains to investigate only the cases where
the rulings are nonnull.
92
6. Ruled Weingarten Surfaces
Case 1 : M ∈ M11 . Then
H=
1 εJt2 − εQ′ t − Q2 (λ + J)
, and
¯
¯
2
¯Q2 − εt2 ¯3/2
1
HII =
2
−
Note that
2J 4
t + ε(5J + 2λ) t2 + 3εQ′ t + Q2 (λ − 3J)
Q2
.
¯
¯
¯Q2 − εt2 ¯3/2
W := Hs (HII )t − Ht (HII )s =
7
X
1
Ai ti ,
2Q3 (Q2 − εt2 )4
i=0
where the coefficients Ai = Ai (s) are real-valued functions of s. Suppose
that W = 0. The vanishing of the coefficients A0 , A1 and A7 implies
Q′ (3QJ ′ − 3Q′ J + Qλ′ − Q′ λ) = 0,
QJJ ′ − Q′ J 2 + 4QJ ′ λ − 4Q′ Jλ + Qλλ′ − Q′ λ2 = 0,
2J(QJ ′ − Q′ J) = 0.
(6.1)
(6.2)
(6.3)
First, assume that Q′ = 0. Thus Q is constant. It follows from (6.3)
that (J 2 )′ = 0, so J 2 is constant or, equivalently, J is constant. We infer
from (6.2) that (λ2 )′ = 0, so λ is constant. In this case,
W = 0 ⇐⇒ Q, J, F are constants,
which corresponds to the fact that M is a helicoidal ruled surface. Moreover, if M is minimal, then it is a Lorentzian helicoid.
From now on, assume that Q′ 6= 0. Since 3QJ ′ −3Q′ J +Qλ′ −Q′ λ = 0,
¡
¢′
it follows that (3J/Q + λ/Q)′ = 0. From (6.3), (J/Q)2 = 0. This
implies that J/Q is constant, so is λ/Q.
Put
a :=
J
≡ constant
Q
so
H=
and b :=
λ
≡ constant,
Q
1 εaQt2 − εQ′ t − (b + a)Q3
¯
¯
2
¯Q2 − εt2 ¯3/2
(6.4)
6.3 Ruled Weingarten Surfaces in E31
and
HII =
Thus
W =−
1
2
−
93
2a 4
t + ε(5a + 2b)Q t2 + 3εQ′ t + (b − 3a)Q3
Q
.
¯
¯
¯Q2 − εt2 ¯3/2
³
¢
¡ ¢2 ´ ¡ 6
at + εbQ2 t4 − (a + b)Q4 t2
QQ′′ − 2 Q′
2Q2 (Q2 − εt2 )4
= 0.
Observe that at6 + εbQ2 t4 − (a + b)Q4 t2 = 0 ⇐⇒ a = b = 0.
Subcase 1.1 : a = b = 0. Thus J = F = 0, so α′′ = ε(F − QJ)β ′ = 0,
and so the striction line α is a straight line. Since hα′ , βi = F = 0, β is
orthogonal to α. Therefore, M is a conoid.
Subcase 1.2 : a2 + b2 6= 0. Thus
¡ ¢2
QQ′′ − 2 Q′ = 0.
The general solution of this differential equation is
Q=
c1
, where c0 , c1 are constants,
s + c0
and we can put c0 = 0 by changing of the variable s (if necessary). Thus,
by (6.4),
J=
c2
,
s
F =
c3
, where c2 , c3 are constants with c22 + c23 6= 0.
s2
Case 2 : M ∈ M01 . From (5.26) and (5.27), we obtain that
∂(H, HII )
=0
∂(s, t)
⇐⇒ 3Q2 Q′′ R′ + 2(Q′ )3 R − Q2 Q′ R′′ − 2Q(Q′ )2 R′ − 2QQ′ Q′′ R = 0
⇐⇒ either Q is constant and R is arbitrary, or Q is arbitrary and R is
given by
R = aQ + bQ
for some constants a, b .
Z
(Q′ )3
ds
Q4
94
6. Ruled Weingarten Surfaces
To verify the above conclusion: Since Q never vanishes, it follows that
R(s) = f (s)Q(s), where f (s) = R(s)/Q(s) is a smooth function. Replace
R by f Q in
3Q2 Q′′ R′ + 2(Q′ )3 R − Q2 Q′ R′′ − 2Q(Q′ )2 R′ − 2QQ′ Q′′ R = 0 .
(6.5)
Hence we have
f ′′ QQ′ + 4f ′ (Q′ )2 − 3f ′ QQ′′ = 0 .
(6.6)
Evidently, (6.6) holds provided that f ′ or Q′ is identically zero. Assume
additionally that f ′ 6= 0 and Q′ 6= 0; thus
f ′′
Q′′
Q′
=
3
−
4
.
f′
Q′
Q
This implies that f ′ = b(Q′ )3 /Q4 for some constant b. Therefore,
Z
(Q′ )3
f =a+b
Q4
for some constants a and b.
The proof is complete.
Theorem 6.15 (Classification of ruled {KII , H}-W-surfaces in
E31 ). let M : x(s, t) = α(s) + tβ(s) be a nondevelopable ruled surface in
E31 . Then M is a ruled {KII , H}-W-surface if and only if it satisfies one
of the following properties:
1. M ∈ M11 , and it is one of the following surfaces:
(1.1) a helicoidal ruled surface;
(1.2) a Lorentzian conoid;
and
(1.3) a surface with invariants
Q(s) =
c1
,
s
J(s) =
c2
,
s
F (s) =
c3
,
s2
where c1 , c2 , c3 are constants with c1 6= 0 and c22 + c23 6= 0, and
s is an arc-length parameter of the regular director curve β.
6.3 Ruled Weingarten Surfaces in E31
95
Moreover, if M is minimal, then it is a Lorentzian helicoid.
2. M ∈ M01 is a surface with nonvanishing invariants Q(s), R(s) such
that either Q is constant and R is arbitrary, or Q is arbitrary and
R is given by
R = aQ + bQ
for some constants a, b .
Z
(Q′ )3
ds
Q4
3. M ∈ M0 .
Proof. Since all nondevelopable ruled surfaces in E31 with null rulings are
ruled {KII , H}-W-surfaces, it remains to investigate only the cases where
the rulings are nonnull.
Case 1 : M ∈ M11 . Then
H=
Note that
1 εJt2 − εQ′ t − Q2 (λ + J)
, and
¯
¯
2
¯Q2 − εt2 ¯3/2
J 4
t + ε(λ − 2J) t2 + 2εQ′ t + Q2 (λ + J)
1 Q2
.
KII =
¯
¯
2
¯Q2 − εt2 ¯3/2
W := (KII )s Ht − (KII )t Hs =
7
X
1
Ai ti ,
4Q3 (Q2 − εt2 )4
i=0
where the coefficients Ai = Ai (s) are real-valued functions of s. Suppose
that W = 0. The vanishing of the coefficients A0 , A1 and A7 implies
Q′ (Qλ′ − Q′ λ + QJ ′ − Q′ J) = 0,
(λ − J)(Qλ′ − Q′ λ + QJ ′ − Q′ J) = 0,
2J(QJ ′ − Q′ J) = 0.
(6.7)
(6.8)
(6.9)
First, assume that Q′ = 0. Thus Q is constant. It follows from (6.9)
that (J 2 )′ = 0; hence J is constant. We infer from (6.8) that (λ−J)λ′ = 0,
96
6. Ruled Weingarten Surfaces
¡
¢′
so (λ − J)2 = 0, and so λ − J is constant. Consequently, λ is constant
as well. In this case,
W = 0 ⇐⇒ Q, J, F are constants.
Therefore, M is a helicoidal ruled surface. Moreover, if M is minimal,
then it is a Lorentzian helicoid.
From now on, assume that Q′ 6= 0. It follows from (6.7) and (6.9)
¡
¢′
that (λ/Q + J/Q)′ = 0 and (J/Q)2 = 0, respectively. Thus J/Q and
λ/Q are constants.
Put
a :=
J
≡ constant
Q
so
H=
and
and b :=
λ
≡ constant,
Q
(6.10)
1 εaQt2 − εQ′ t − (b + a)Q3
¯
¯
2
¯Q2 − εt2 ¯3/2
a 4
t + ε(b − 2a)Q t2 + 2εQ′ t + (a + b)Q3
1 Q
KII =
.
¯
¯
2
¯Q2 − εt2 ¯3/2
Since W = 0, it follows that
³
¢
¡ ¢2 ´ ¡ 6
at − ε(4a + b)Q2 t4 + (3a + b)Q4 t2
QQ′′ − 2 Q′
4Q2 (Q2 − εt2 )4
= 0.
Observe that at6 − ε(4a + b)Q2 t4 + (3a + b)Q4 t2 = 0 ⇐⇒ a = b = 0.
Subcase 1.1 : a = b = 0. Thus J = F = 0, so α′′ = ε(F − QJ)β ′ = 0,
and so the striction line α is a straight line. Since hα′ , βi = F = 0, β is
orthogonal to α. Therefore, M is a conoid.
Subcase 1.2 : a2 + b2 6= 0. Thus
¡ ¢2
QQ′′ − 2 Q′ = 0 .
The general solution of this differential equation is
6.3 Ruled Weingarten Surfaces in E31
97
c1
, where c0 , c1 are constants,
s + c0
and we can put c0 = 0 by changing of the variable s (if necessary). Thus,
Q=
by (6.10),
J=
c2
,
s
F =
c3
, where c2 , c3 are constants with c22 + c23 6= 0.
s2
Case 2 : M ∈ M01 . From (5.26) and (5.25), we obtain that
∂(KII , H)
=0
∂(s, t)
⇐⇒ 3Q2 Q′′ R′ + 2(Q′ )3 R − Q2 Q′ R′′ − 2Q(Q′ )2 R′ − 2QQ′ Q′′ R = 0
⇐⇒ either Q is constant and R is arbitrary, or Q is arbitrary and R is
given by
R = aQ + bQ
for some constants a, b .
Z
(Q′ )3
ds
Q4
Theorem 6.16 (Classification of ruled {KII , HII }-W-surfaces in
E31 ). let M : x(s, t) = α(s) + tβ(s) be a nondevelopable ruled surface in
E31 . Then M is a ruled {KII , HII }-W-surface if and only if it satisfies one
of the following properties:
1. M ∈ M11 , and it is one of the following surfaces:
(1.1) a helicoidal ruled surface;
(1.2) a Lorentzian conoid;
and
(1.3) a surface with invariants
Q(s) =
c1
,
s
J(s) =
c2
,
s
F (s) =
c3
,
s2
where c1 , c2 , c3 are constants with c1 6= 0 and c22 + c23 6= 0, and
s is an arc-length parameter of the regular director curve β.
Moreover, if M is minimal, then it is a Lorentzian helicoid.
98
6. Ruled Weingarten Surfaces
2. M ∈ M01 is a surface with nonvanishing invariants Q(s), R(s) such
that either Q is constant and R is arbitrary, or Q is arbitrary and
R is given by
R = aQ + bQ
Z
(Q′ )3
ds
Q4
for some constants a, b .
3. M ∈ M0 .
Proof. Since all nondevelopable ruled surfaces in E31 with null rulings
are ruled {KII , HII }-W-surfaces, it remains to investigate only the cases
where the rulings are nonnull.
Case 1 : M ∈ M11 . Then
J 4
t + ε(λ − 2J) t2 + 2εQ′ t + Q2 (λ + J)
1 Q2
KII =
, and
¯
¯
2
¯Q2 − εt2 ¯3/2
HII =
Note that
1
2
−
2J 4
t + ε(5J + 2λ) t2 + 3εQ′ t + Q2 (λ − 3J)
Q2
.
¯
¯
¯Q2 − εt2 ¯3/2
7
X
1
W := (KII )s (HII )t − (KII )t (HII )s =
Ai ti ,
4Q3 (Q2 − εt2 )4
i=0
where the coefficients Ai = Ai (s) are real-valued functions of s. Suppose
that W = 0. The vanishing of the coefficients A0 , A1 , A3 and A7 implies
Q′ (9QJ ′ − 9Q′ J + Qλ′ − Q′ λ) = 0, (6.11)
QJJ ′ − Q′ J 2 − Qλλ′ − QJλ′ − 11QJ ′ λ + 12Q′ Jλ + Q′ λ2 = 0, (6.12)
18Q′ Jλ + Q′ λ2 − QJλ′ + 3QJJ ′ − 3Q′ J 2 − 17QJ ′ λ − Qλλ′ = 0, (6.13)
2QJJ ′ + 4QJλ′ − 2Q′ J 2 + 4QJ ′ λ − 8Q′ Jλ = 0. (6.14)
6.3 Ruled Weingarten Surfaces in E31
99
First, assume that Q′ = 0. Thus Q is constant. The equations (6.12),
(6.13) and (6.14) become
JJ ′ − λλ′ − Jλ′ − 11J ′ λ = 0,
(6.15)
−Jλ′ + 3JJ ′ − 17J ′ λ − λλ′ = 0,
(6.16)
2JJ ′ + 4Jλ′ + 4J ′ λ = 0.
(6.17)
Subtracting (6.16) from (6.15), we get
2(3λ − J)J ′ = 0.
(6.18)
If J ′ = 0, then J is constant; we deduce from (6.17) that Jλ′ = 0, so
λλ′ = 0 in (6.15), and so λ is constant. On the other hand, if 3λ − J = 0,
then J = 3λ and J ′ = 3λ′ ; the equation (6.17) becomes 21(λ2 )′ = 0;
hence λ is constant, and so is J. In this case,
W = 0 ⇐⇒ Q, J, F are constants.
Therefore, M is a helicoidal ruled surface. Moreover, if M is minimal,
then it is a Lorentzian helicoid.
From now on, assume that Q′ 6= 0. The equations (6.11) and (6.14)
are equivalent to
µ
9J
λ
+
Q
Q
¶′
= 0 and
Ã
4J λ
+
QQ
µ
J
Q
¶2 !′
= 0,
respectively. From these last two equations, we deduce that J/Q and
λ/Q are constants.
Put
a :=
so
J
≡ constant
Q
and b :=
λ
≡ constant,
Q
a 4
t + ε(b − 2a)Q t2 + 2εQ′ t + (a + b)Q3
1 Q
KII =
¯
¯
2
¯Q2 − εt2 ¯3/2
(6.19)
100
6. Ruled Weingarten Surfaces
and
HII =
1
2
−
2a 4
t + ε(5a + 2b)Q t2 + 3εQ′ t + (b − 3a)Q3
Q
.
¯
¯
¯Q2 − εt2 ¯3/2
Since W = 0, it follows that
³
¢
¡ ¢2 ´ ¡ 6
7at + ε(b − 12a)Q2 t4 + (5a − b)Q4 t2
QQ′′ − 2 Q′
4Q2 (Q2 − εt2 )4
= 0.
Observe that 7at6 + ε(b − 12a)Q2 t4 + (5a − b)Q4 t2 = 0 ⇐⇒ a = b = 0.
Subcase 1.1 : a = b = 0. Thus J = F = 0, so α′′ = ε(F − QJ)β ′ = 0,
and so the striction line α is a straight line. Since hα′ , βi = F = 0, β is
orthogonal to α. Therefore, M is a conoid.
Subcase 1.2 : a2 + b2 6= 0. Thus
¡ ¢2
QQ′′ − 2 Q′ = 0.
The general solution of this differential equation is
Q=
c1
, where c0 , c1 are constants,
s + c0
and we can put c0 = 0 by changing of the variable s (if necessary). Thus,
by (6.19),
J=
c2
,
s
F =
c3
, where c2 , c3 are constants with c22 + c23 6= 0.
s2
Case 2 : M ∈ M01 . From (5.25) and (5.27), we obtain that
∂(KII , HII )
=0
∂(s, t)
⇐⇒ 3Q2 Q′′ R′ + 2(Q′ )3 R − Q2 Q′ R′′ − 2Q(Q′ )2 R′ − 2QQ′ Q′′ R = 0
⇐⇒ either Q is constant and R is arbitrary, or Q is arbitrary and R is
given by
R = aQ + bQ
for some constants a, b .
Z
(Q′ )3
ds
Q4
6.3 Ruled Weingarten Surfaces in E31
101
Note 6.17. One can observe that the classifications of nondevelopable
ruled {KII , H}-, {KII , HII }- and {H, HII }-W-surfaces of class M01 are
the same. This is not a surprise because the curvatures KII , H, HII in
this case satisfy HII − H = 2KII .
Theorem 6.18 (Classification of ruled {K, KII }-W-surfaces in
E31 ). Let M : x(s, t) = α(s) + tβ(s) be a nondevelopable ruled surface in
E31 . Then M is a ruled {K, KII }-W-surface if and only if it satisfies one
of the following properties:
1. M ∈ M11 is a helicoidal ruled surface. Moreover, if M is minimal,
then it is a Lorentzian helicoid.
2. M ∈ M01 is a surface with nonvanishing invariants Q(s), R(s) such
that Q is arbitrary and R is given by
Z p
R = aQ + bQ
|Q| ds
for some constants a, b .
3. M ∈ M0 .
Proof. Since all nondevelopable ruled surfaces in E31 with null rulings are
ruled {K, KII }-W-surfaces, it remains to investigate only the cases where
the rulings are nonnull.
Case 1 : M ∈ M11 . Thus
Q2
, and
(Q2 − εt2 )2
J 4
t + ε(λ − 2J) t2 + 2εQ′ t + Q2 (λ + J)
1 Q2
.
KII =
¯
¯
2
¯Q2 − εt2 ¯3/2
K=
Note that
W := Ks (KII )t − Kt (KII )s =
1
¯
¯9/2
Q¯Q2 − εt2 ¯
5
X
i=0
Ai ti ,
102
6. Ruled Weingarten Surfaces
where the coefficients Ai = Ai (s) are real-valued functions of s. Suppose
that W = 0. The vanishing of the coefficients A0 , A3 and A5 implies
(Q′ )2 = 0,
4QJ ′ − 2Qλ′ − 6Q′ J + Q′ λ = 0,
and
2QJ ′ − 3Q′ J = 0.
Clearly, Q, J and F are constants; thus M is a helicoidal ruled surface.
Moreover, if M is minimal, then it is a Lorentzian helicoid.
Case 2 : M ∈ M01 . From (5.24) and (5.25), we obtain that
∂(K, KII )
=0
∂(s, t)
⇐⇒ 2QQ′′ R + 5QQ′ R′ − 2Q2 R′′ − 5(Q′ )2 R = 0
⇐⇒ Q is arbitrary and R is given by
Z p
|Q| ds
R = aQ + bQ
for some constants a, b .
In 1999, Dillen and Kühnel [10] classified nondevelopable ruled {K, H}-
W-surface in E31 . We include their results here with a detailed proof.
Theorem 6.19 (Classification of ruled {K, H}-W-surfaces in E13 ).
([10], Theorem 2). Let M : x(s, t) = α(s) + tβ(s) be a nondevelopable
ruled surface in E31 . Then M is a ruled {K, H}-W-surface if and only if
it satisfies one of the following properties:
1. M ∈ M11 is a helicoidal ruled surface. Moreover, if M is minimal,
then it is a Lorentzian helicoid.
2. M ∈ M01 is a helicoidal ruled surface. Moreover, if M is minimal,
then it is a conjugate of Enneper’s surface of the 2nd kind.
3. M ∈ M0 .
6.3 Ruled Weingarten Surfaces in E31
103
Proof. Since all nondevelopable ruled surfaces in E31 with null rulings are
ruled {K, H}-W-surfaces, it remains to investigate only the cases where
the rulings are nonnull.
Case 1 : M ∈ M11 . From (5.19) and (5.21), we have
1
W := Ks Ht − Kt Hs = ¯
¯
¯Q2 − εt2 ¯9/2
3
X
Ai ti ,
i=0
where the coefficients Ai = Ai (s) are real-valued functions of s. Suppose
that W = 0. The vanishing of the coefficients A0 , A1 and A3 implies
(Q′ )2 = 0,
2QJ ′ + 2Qλ′ + Q′ λ − Q′ J = 0,
and
2QJ ′ − Q′ J = 0.
Clearly, Q, J and F are constants. By Lemma 5.14, M is a helicoidal
ruled surface. Moreover, if M is minimal, then it is a Lorentzian helicoid.
Case 2 : M ∈ M01 . From (5.24) and (5.26), we have
4H = ±
³ R ´′ Q
Q′
1/4
K
+
K 3/4 .
Q |Q|3/2
Q|Q|1/2
(6.20)
We claim that M is a {K, H}-W-surface if and only if the two coeffi-
cients in (6.20) are constants or, equivalently,
Q′
= C1
|Q|3/2
and
³ R ´′
Q
1
= C2
|Q|1/2
(6.21)
for some constants C1 and C2 . The “sufficiency” part is obvious. Conversely, suppose that M is a {K, H}-W-surface. Since ∂K/∂t 6= 0, it
follows from Theorem A.7 that (locally) H = φ(K) for some C1 function
φ. Now let Y = K 1/4 and regard Y and s as independent variables, so
(6.20) becomes
4φ(Y 4 ) = ±
³ R ´′ Q
Q′
Y
+
Y 3.
Q |Q|3/2
Q|Q|1/2
(6.22)
104
6. Ruled Weingarten Surfaces
Differentiate both sides of (6.22) with respect to s, we get
µ
±
Q′
Q|Q|1/2
¶′
Y +
µ³ ´
R ′
Q
Q
|Q|3/2
¶′
Y 3 = 0.
Therefore, the two coefficients of this equation must be zero and we thus
have the claim.
The solutions to (6.21)1 are Q is constant if C1 = 0, and Q =
4
± C 2 (s+C
1
3)
2
otherwise, where C3 is another constant. In the latter case,
we may assume, after translation in the parameter s that C3 = 0. Hence
either Q = A or Q =
A
s2
for some nonzero constant A.
Subcase: 2.1 : Q = A. Thus
R
Q
= Ds + C for some constants D and C.
After translation, M is parametrized by
x(s, t) =
1
2


=


D 2
s
2
 D s2
2
1+
s2
2
s2
2
s
 
s
 

+ t s
+ Cs + As −

2
1
−As
 3

 


2
s
D
−D
− s2
s
+
s
2

  2 
 A  33
2

 −  s − s +  0  .
1 − s2 s
0

 
 2 3

D
C
C
2
−s
1
s
t+ 2s+ 2
−2
+ Cs − As −

A 3
3s 
A 3
3s 
This shows that M is obtained by a cubic screw motion applied to the
straight line passing through (− D
2 , 0, 0) in the direction of (0, 0, 1) and
C
then translated by ( D
2 , 0, − 2 ). Therefore, M is a helicoidal ruled surface.
A
.
s2
R
Q
= D ln |s|+C for some constants D and C.
¡
¢
After translation, we can assume that x(s, t) = x1 (s, t), x2 (s, t), x3 (s, t)
Subcase 2.2 : Q =
Then
= α(s) + tβ(s) satisfies
A
,
s
x1 + x2 = D(s ln |s| − s) + 2st + (C − A)s, and
³
D C −A
D´
ln |A| −
+
.
x3 = −A ln |s| + t + A +
2
2
2
x1 − x2 =
6.3 Ruled Weingarten Surfaces in E31
105
In order to show that the surface M in this case is a Nomizu–Sasaki
surface, we need to use another base curve α
e(s) = α(s) −
C−A
2 β(s),
so
the new parametrization satisfies
A
,
s
x1 + x2 = D(s ln |s| − s) + 2st, and
³
D´
D
x3 = −A ln |s| + t + A +
ln |A| − .
2
2
x1 − x2 =
Consequently, M is given by
x3 =
¢ ¡
D¢
1 ¡ 2
x1 − x22 + A +
ln |x1 − x2 |,
2A
2
which is a Nomizu–Sasaki surface, as required.
The proof is complete.
Theorem 6.20 (Classification of ruled {K, HII }-W-surfaces in
E31 ). Let M : x(s, t) = α(s) + tβ(s) be a nondevelopable ruled surface in
E31 . Then M is a ruled {K, HII }-W-surface if and only if it satisfies one
of the following properties:
1. M ∈ M11 is a helicoidal ruled surface. Moreover, if M is minimal,
then it is a Lorentzian helicoid.
2. M ∈ M01 is a helicoidal ruled surface. Moreover, if M is minimal,
then it is a conjugate of Enneper’s surface of the 2nd kind.
3. M ∈ M0 .
Proof. Since all nondevelopable ruled surfaces in E31 with null rulings are
ruled {K, HII }-W-surfaces, it remains to investigate only the cases where
the rulings are nonnull.
Case 1 : M ∈ M11 . From (5.19) and (5.22), we have
W := Ks (HII )t − Kt (HII )s =
1
¯
¯9/2
Q¯Q2 − εt2 ¯
5
X
i=0
Ai ti ,
106
6. Ruled Weingarten Surfaces
where the coefficients Ai = Ai (s) are real-valued functions of s. Suppose
that W = 0. The vanishing of the coefficients A0 , A3 and A5 implies
(Q′ )2 = 0,
13Q′ J + 2Q′ λ − 10QJ ′ − 4Qλ′ = 0,
and
3Q′ J − 2QJ ′ = 0.
Clearly, Q, J and F are constants. By Lemma 5.14, M is a helicoidal
ruled surface. Moreover, if M is minimal, then it is a Lorentzian helicoid.
Case 2 : M ∈ M01 . From (5.24) and (5.27), we have
4HII = ±
³ R ´′ Q
Q′
1/4
K
−
3
K 3/4 .
Q |Q|3/2
Q|Q|1/2
Hence M is a {K, HII }-W-surface if and only if
³ R ´′ 1
Q′
and
are constants.
Q |Q|1/2
|Q|3/2
By the same argument given in Theorem 6.19, we deduce that the classifications of nondevelopable ruled {K, HII }-W-surfaces of class M01 and of
nondevelopable ruled {K, H}-W-surfaces of class M01 are the same.
6.4
Synopsis
We here summarize the classifications of nondevelopable ruled Weingarten
surfaces in Minkowski 3-space.
Theorem 6.21. (Classification of ruled {A, B}-W-surfaces in E31 ,
where (A, B) ∈ {(KII , H), (KII , HII ), (H, HII )}).
Let (A, B) ∈ {(KII , H), (KII , HII ), (H, HII )}, and let M : x(s, t) = α(s)+
tβ(s) be a nondevelopable ruled surface in E31 . Then M is a ruled {A, B}-
W-surface if and only if it satisfies one of the following properties:
1. M ∈ M11 , and it is one of the following surfaces:
6.4 Synopsis
107
(1.1) a helicoidal ruled surface;
(1.2) a Lorentzian conoid;
and
(1.3) a surface with invariants
Q(s) =
c1
,
s
J(s) =
c2
,
s
F (s) =
c3
,
s2
where c1 , c2 , c3 are constants with c1 6= 0 and c22 + c23 6= 0, and
s is an arc-length parameter of the regular director curve β.
Moreover, if M is minimal, then it is a Lorentzian helicoid.
2. M ∈ M01 is a surface with nonvanishing invariants Q(s), R(s) such
that either Q is constant and R is arbitrary, or Q is arbitrary and
R is given by
R = aQ + bQ
for some constants a, b .
Z
(Q′ )3
ds
Q4
3. M ∈ M0 .
Theorem 6.22. (Classification of ruled {K, KII }-W-surfaces in
E31 ).
Let M : x(s, t) = α(s) + tβ(s) be a nondevelopable ruled surface in E31 .
Then M is a ruled {K, KII }-W-surface if and only if it satisfies one of
the following properties:
1. M ∈ M11 is a helicoidal ruled surface. Moreover, if M is minimal,
then it is a Lorentzian helicoid.
2. M ∈ M01 is a surface with nonvanishing invariants Q(s), R(s) such
that Q is arbitrary and R is given by
Z p
|Q| ds
R = aQ + bQ
for some constants a, b .
108
6. Ruled Weingarten Surfaces
3. M ∈ M0 .
Theorem 6.23. (Classification of ruled {A, B}-W-surfaces in E31 ,
where (A, B) ∈ {(K, H), (K, HII )}).
Let (A, B) ∈ {(K, H), (K, HII )}, and let M : x(s, t) = α(s) + tβ(s) be a
nondevelopable ruled surface in E31 . Then M is a ruled {A, B}-W-surface
if and only if it satisfies one of the following properties:
1. M ∈ M11 is a helicoidal ruled surface. Moreover, if M is minimal,
then it is a Lorentzian helicoid.
2. M ∈ M01 is a helicoidal ruled surface. Moreover, if M is minimal,
then it is a conjugate of Enneper’s surface of the 2nd kind.
3. M ∈ M0 .
Note 6.24. Although it is difficult to determine functional relations between two curvatures of Weingarten surfaces, there are some situations
that we can find the functional relations easily. For instance, it follows
from the proofs and the results of Theorems 6.19 and 6.20 that the curvatures K, H and HII satisfy
(i) H = AK 1/4 + BK 3/4 + CK 1/2 on nondevelopable ruled {K, H}-Wsurfaces in E31
and
e −1/4 + BK
e 1/4 + CK
e 3/4 + DK
e 1/2 on nondevelopable ruled
(ii) HII = AK
{K, HII }-W-surfaces in E31
e B, B,
e C, C
e and D.
e
for some constants A, A,
Chapter
7
Ruled Linear Weingarten Surfaces
This chapter is devoted to the study of nondevelopable ruled surfaces on
which the linear combination aKII + bH + cHII + dK is constant along
each ruling, where a, b, c, d are constants such that a2 + b2 + c2 6= 0.
7.1
Ruled Linear Weingarten Surfaces in E3
In 1992, D. E. Blair and Th. Koufogiorgos [3] studied a nondevelopable
ruled surface in E3 such that the linear combination aKII + bH, where a
and b are real constants with 2a + b 6= 0, is constant along every ruling;
such a surface is a helicoid, which is minimal and II-flat. Apart from
Weingarten surfaces, G. Stamou [31] also generalized the result of Blair
and Koufogiorgos by investigating nondevelopable ruled surfaces in E3
such that the linear combination aKII + bH + cHII is constant along
every ruling, where a, b, c are constants with a2 + b2 + c2 6= 0.
We hereby take the Gaussian curvature into account by considering
nondevelopable ruled surfaces in E3 such that aKII + bH + cHII + dK,
where a, b, c, d are constants with a2 + b2 + c2 6= 0, is constant along each
ruling. Because the proof of the following theorem is similar to the one
of Theorem 7.5 below, we just state the theorem without proof. One can
109
110
7. Ruled Linear Weingarten Surfaces
also consult the original work of Stamou.
Theorem 7.1. Let M : x(s, t) = α(s) + tβ(s) be a nondevelopable ruled
surface in E3 , and let a, b, c, d be real constants such that a2 + b2 + c2 6= 0.
Suppose that aKII + bH + cHII + dK is constant along each ruling of M .
Then d = 0 and aKII + bH + cHII + dK = 0. Moreover, M satisfies the
following properties:
1. If 2a + b − 3c 6= 0, then M is an orthogonal ruled surfaces with
nonzero constant parameter of distribution Q.
2. If 2a + b − 3c = 0, then M is a conoidal ruled surface or the relation
F − 2JQ = 0 holds.
The following table gives the relations between KII , H and HII associated with the ruled surfaces mentioned in Theorem 7.1:
2a + b − 3c 6= 0
Conditions for the
Relations between
invariants Q, J, F
KII , H, HII
Q′ = F = 0
2KII + H − HII = 0
In particular, if J = 0,
then KII = H = HII = 0
J =0
2KII − H + HII = 0
In particular, if F = 0,
then 3KII = 6H = −2HII
2a + b − 3c = 0
F − 2JQ = 0
2KII − 7H − HII = 0
The following 3 corollaries follow immediately from the proof of Theorem 7.1.
7.2 Ruled Linear Weingarten Surfaces in E31
111
Corollary 7.2. If M is a nondevelopable ruled surface in E3 such that
aKII + bH, where 2a + b 6= 0, is constant along each ruling, then KII =
H = 0.
Corollary 7.3. If M is a nondevelopable ruled surface in E3 such that
bH + cHII , where b − 3c 6= 0, is constant along each ruling, then H =
HII = 0.
Corollary 7.4. If M is a nondevelopable ruled surface in E3 such that
aKII + cHII , where 2a − 3c 6= 0, is constant along each ruling, then
KII = HII = 0.
7.2
Ruled Linear Weingarten Surfaces in E31
In this section, we will categorize nondevelopable ruled surfaces in E31
with nonnull rulings such that aKII + bH + cHII + dK is constant along
every ruling, where a, b, c, d are constants with a2 + b2 + c2 6= 0.
Theorem 7.5. Let M : x(s, t) = α(s) + tβ(s) be a nondevelopable ruled
surface in E31 with nonnull rulings, and let a, b, c, d be constants such that
a2 +b2 +c2 6= 0. Suppose that aKII +bH +cHII +dK is constant along each
ruling of M . Then d = 0 and aKII + bH + cHII + dK = 0. Furthermore,
M satisfies the following properties:
1. Suppose that β ′ is nowhere null.
(1.1) If 2a − b + 3c 6= 0, then M is an orthogonal ruled surface with
nonzero constant invariant Q.
(1.2) If 2a − b + 3c = 0, then M is a conoidal ruled surface or the
relation 2JQ + F = 0 holds.
2. Suppose that β ′ is null everywhere.
(2.1) If 2a−b+3c 6= 0, then M is II-flat. Moreover, M is a conjugate
of Enneper’s surface of the 2nd kind provided that b + c 6= 0.
112
7. Ruled Linear Weingarten Surfaces
(2.2) If 2a − b + 3c = 0, then the invariant Q is a nonzero constant
or a = 2b = −2c (6= 0).
p
|Q2 − εt2 | is
¡
a rational function of the variable t if and only if d = 0. Since aKII +
¢
bH + cHII + dK t = 0, it follows that d = 0 and that
Proof. Suppose first that β ′ is nowhere null. Note that d
(a − 2c) J
= 0,
(7.1)
(3a + b − 5c) QJ + (a + 2c) F
= 0,
(7.2)
(−3a − 2b + 4c) QJ + (4a − 3b + 5c) F
= 0,
(7.3)
(−a − b + c) QJ + (5a − 3b + 7c) F
= 0,
and
(2a − b + 3c) Q′ = 0.
(7.4)
(7.5)
Case 1 : 2a − b + 3c 6= 0. From (7.5), we get Q′ = 0.
Subcase 1.1 : J = 0. By computing 3 × (7.2) + (7.3) + (7.4), we have
6(2a − b + 3c) F = 0; hence F = 0.
Subcase 1.2 : J 6= 0. It follows from (7.1) that a − 2c = 0, so (7.2), (7.3)
and (7.4) become
(b + c) QJ + 4c F
= 0,
(−2b − 2c) QJ + (−3b + 13c) F
= 0,
(−b − c) QJ + (−3b + 17c) F
= 0.
(7.6)
and
(7.7)
(7.8)
By calculating 2 × (7.8) − (7.7), we get −3(b − 7c) F = 0. If b − 7c = 0,
then 2a − b + 3c = 2(a − 2c) − (b − 7c) = 0, which is a contradiction.
Thus b − 7c 6= 0, so F = 0. Furthermore, a = −2b = 2c (6= 0) and (7.6)
is satisfied.
In this case, M is an orthogonal ruled surface.
7.2 Ruled Linear Weingarten Surfaces in E31
113
Case 2 : 2a − b + 3c = 0. Then (7.2), (7.3) and (7.4) become
(5a − 2c) QJ + (a + 2c) F
= 0,
(−7a − 2c) QJ + (−2a − 4c) F
= 0,
(−3a − 2c) QJ + (−a − 2c) F
= 0.
(7.9)
and
(7.10)
(7.11)
Subcase 2.1 : J = 0. We deduce that (a + 2c) F = 0; hence a + 2c = 0
or F = 0. In the first case, a = 2b = −2c (6= 0). In the second case,
M is a conoid with 3KII = −6H = 2HII ; it follows that aKII + bH +
¡
¢
¡
¢
cHII = a KII + 2H + c 3H + HII = 0. In either case, we finally get
2KII + H − HII = 0.
Subcase 2.2 : J 6= 0. By (7.1), we get a − 2c = 0, so 7a = 2b = 14c (6= 0).
In addition, (7.9), (7.10) and (7.11) imply that 2JQ + F = 0.
From now on, suppose that β ′ is null everywhere. We note that
p
d |R + 2Qt| is a rational function of the parameter t if and only if d = 0.
¡
¢
Since aKII + bH + cHII + dK t = 0, it follows that d = 0 and that
(6a − 3b + 9c)QR′ + (−6a + 4b − 8c)Q′ R = 0,
(6a − 3b + 9c)QR′ + (−6a + 5b − 7c)Q′ R = 0, and
(b + c)Q′ = 0.
(7.12)
(7.13)
(7.14)
Case 1: 2a − b + 3c 6= 0. We have 2 subcases.
Subcase 1.1: Q′ = 0. Since 2a − b + 3c 6= 0 and Q 6= 0, it follows from
(7.12) that R′ = 0. Hence KII = H = HII = 0.
Subcase 1.2: Q′ 6= 0. It follows from (7.14) that b + c = 0. Thus (7.12)
and (7.13) become
(6a + 12c)(QR′ − Q′ R) = 0.
But 2a − b + 3c 6= 0 and −b = c, so 6a + 12c 6= 0. We infer that KII = 0.
Case 2: 2a − b + 3c = 0. Thus (7.12), (7.13) and (7.14) imply (b +
c)Q′ = 0. Hence the invariant Q is a nonzero constant or a = 2b =
−2c (6= 0).
114
7. Ruled Linear Weingarten Surfaces
In each case, we always get
aKII + bH + cHII + dK = 0.
The following table gives the relations between KII , H and HII associated with the ruled surfaces mentioned in Theorem 7.5:
β and β ′ are
Conditions for the
Relations between
nowhere null
invariants Q, J, F
KII , H, HII
2a − b + 3c 6= 0
Q′ = F = 0
In particular, if J = 0,
then KII = H = HII = 0
2KII + H − HII = 0
J =0
In particular, if F = 0,
then 3KII = −6H = 2HII
2a − b + 3c = 0
2JQ + F = 0
β is nowhere null, but Conditions for the
β′
2KII − H + HII = 0
is null everywhere
invariants Q, R
2a − b + 3c 6= 0
RQ′
2KII + 7H + HII = 0
Relations between
KII , H, HII
KII = 0
2a − b + 3c = 0
−
R′ Q
=0
In particular, if Q′ = 0,
then KII = H = HII = 0
Q′ = 0
aKII + bH + cHII = 0
Q′ 6= 0
2KII + H − HII = 0
Remark 7.6. Since all surfaces in Examples 6.3–6.6 are nondevelopable,
II-flat, minimal and II-minimal, it follows that the linear combination
7.2 Ruled Linear Weingarten Surfaces in E31
115
aKII + bH + cHII + dK becomes dK which is not constant along any
ruling unless d = 0.
We hereby take the nondevelopable ruled surfaces with null rulings
into account, so we have the following 3 corollaries:
Corollary 7.7. If M is a nondevelopable ruled surface in E31 (with or
without null rulings) such that aKII +bH, where b(2a−b) 6= 0, is constant
along each ruling, then the equality KII + H = 0 holds on M .
Corollary 7.8. If M is a nondevelopable ruled surface in E31 (with or
without null rulings) such that bH + cHII , where (b + c)(3c − b) 6= 0, is
constant along each ruling, then the equality H − HII = 0 holds on M .
Corollary 7.9. If M is a nondevelopable ruled surface in E31 (with or
without null rulings) such that aKII + cHII , where c(2a + 3c) 6= 0, is
constant along each ruling, then the equality KII + HII = 0 holds on M .
Remark. If the rulings of M in Corollaries 7.7–7.9 are nowhere null, then
KII = H = HII = 0.
Appendix
A
Some Results from Real Analysis
We give some results from real analysis. On functional dependence we
discuss necessary and sufficient conditions for the existence of a functional
relation with detailed proof.
Continuous Additive Functions
Definition A.1. A function f : R → R is said to be additive if f (x+y) =
f (x) + f (y) for all x, y ∈ R.
Proposition A.2. ([32], p. 307). If f : R → R is additive and continuous
on R and if a = f (1), then f (x) = ax for all x ∈ R.
Basic Theorems
Theorem A.3 (Mean Value Theorem). ([13], p. 86). Let Ω be open
in Rn and let f : Ω → R be differentiable on Ω. If Ω contains the line
segment with end points a and a + h, then there is a point c = a + th
with 0 < t < 1 of this line segment such that
f (a + h) − f (a) = Df (c) · h,
117
(A.1)
118
A. Some Results from Real Analysis
where Df (c) is the derivative of f at c.
Theorem A.4 (Inverse Function Theorem). ([28], Theorem 9.24).
Let Ω be an open subset of Rn and let f : Ω → Rn be a C1 -mapping.
Suppose that Df (a) is invertible for some a ∈ Ω and that b = f (a).
Then:
1. There exist open sets U and V in Rn such that a ∈ U , b ∈ V , and
¯
f ¯ : U → V is bijective.
U
¯
2. If g : V → U is the inverse of f ¯U , then g is also of class C1 .
Theorem A.5 (Implicit Function Theorem). ([19], pp. 3–4). Let
U1 ⊆ Rk and U2 ⊆ Rm be open sets and let
f : U1 × U2 → Rm
be a C1 -mapping. Let (x, y) = (x1 , . . . , xk , y1 , . . . , ym ) denote the standard coordinates on U1 × U2 . Suppose that (a, b) is a point in U1 × U2
such that f (a, b) = 0 and that the matrix
¶
µ
∂fi
(a, b)
∂yj
1≤i,j≤m
is invertible. Then there are neighborhoods V1 ⊆ U1 of a, V2 ⊆ U2 of b,
and a C1 -mapping
g : V1 → V2
such that for all (x, y) ∈ V1 × V2 , the implicit equation f (x, y) = 0 holds
if and only if the explicit equation y = g(x) is satisfied.
Functional Dependence
Definition A.6. Two functions f and g are said to be functionally dependent on a set Ω if there is a differentiable real-valued function F of
two real variables such that
A. Some Results from Real Analysis
(i) F itself is not identically zero on any open set,
¡
¢
(ii) F f (p), g(p) = 0
119
and
for all p ∈ Ω.
The function F above is called a functional relation between f and g
on Ω. If F is of class Cr (r ≥ 0), then F is termed a functional relation
of class Cr between f and g on Ω.
Theorem A.7. Let T be a C1 -mapping from a connected open set Ω ⊆ R2
into R2 described by
u = f (x, y)
and
v = g(x, y) .
(A.2)
Suppose that the derivative DT has constant rank either 0 or 1 on Ω.
Then at each point p ∈ Ω, there is a neighborhood Up of p on which
there exists a functional relation of class C1 between u and v such that
one of the functions u and v can be used to express the other. In detail:
1. Suppose that DT has rank 0 on Ω. Then u and v are constant.
2. Suppose that DT has rank 1 on Ω.
(2.1) If fx2 + fy2 6= 0 at p0 ∈ Ω, then there exist a neighborhood Op0
of p0 and a C1 -function φ such that the equation v = φ(u)
holds on Op0 .
epe
e 0 ∈ Ω, then there exist a neighborhood O
(2.2) If gx2 + gy2 6= 0 at p
0
1
e
e
e 0 and a C -function φ such that the equation u = φ(v)
of p
epe .
holds on O
0
R2 ,
Conversely, if u and v are functionally dependent on an open set O ⊆
then
∂(u, v)
(p) = 0
∂(x, y)
for all p ∈ O.
120
A. Some Results from Real Analysis
Proof. If DT has rank 0 on Ω, then it follows immediately from the mean
value theorem that u and v are constant. From now on suppose that DT
has rank 1 on Ω.
To prove (2.1), we may assume without loss of generality that fx 6= 0
at p0 = (x0 , y0 ) ∈ Ω. Define F : Ω × R → R by F (x, y, u
b) = f (x, y) − u
b
for all (x, y, u
b) ∈ Ω × R. Then F is a C1 -function with F (x0 , y0 , u
b0 ) = 0,
where u
b0 = u(p0 ) = f (x0 , y0 ). Note that Fx (x0 , y0 , u
b0 ) = fx (x0 , y0 ),
so Fx (x0 , y0 , u
b0 ) 6= 0.
The implicit function theorem says that near
(x0 , y0 , u
b0 ) the equation F (x, y, u
b) = 0 can be solved for x. More pre-
cisely, we can find open intervals I1 , I2 , I3 in R such that
(i) x0 ∈ I1 , y0 ∈ I2 , u
b0 ∈ I3 , I1 × I2 ⊆ Ω,
and
(ii) there exists a C1 -function K : I2 × I3 → I1 such that for all
(x, y, u
b) ∈ I1 × I2 × I3 , the implicit equation F (x, y, u
b) = 0 holds if
and only if the explicit equation x = K(y, u
b) is satisfied.
Put Op0 = I1 × I2 ; thus Op0 is a neighborhood of p0 . From v =
g(x, y), we define a function G : I2 × I3 → R by
¡
¢
G(y, u
b) = g K(y, u
b), y
(A.3)
for all (y, u
b) ∈ I2 × I3 . Evidently, G is of class C1 . Observe that
(A.4)
¢
¡
on I1 × I2 × I3 ∩ F −1 ({0}) . It follows from (A.3) and (A.4) that
(A.5)
¡
¢
f K(y, u
b), y − u
b=0
Gy = gx Ky + gy
and
0 = fx Ky + fy
¡
¢
on I1 × I2 × I3 ∩ F −1 ({0}) . Solving the second equation, we have
Ky
= −
fy
fx
(A.6)
¢
¡
on I1 × I2 × I3 ∩ F −1 ({0}) . Since DT has rank 1, we infer that
³ f ´
fx gy − fy gx
y
Gy = gx −
+ gy =
=0
(A.7)
fx
fx
A. Some Results from Real Analysis
121
¢
¡
on I1 × I2 × I3 ∩ F −1 ({0}) . This shows that G does not depend on y.
We may therefore write (A.3) in the simpler form v = φ(b
u), where φ is a
C1 -function. In particular, v = φ(u) when u = u
b on Op0 .
(2.2) can be proven similarly.
Conversely, suppose that there is a differentiable functional relation
¡
¢
φ between u and v on an open set O ⊆ R2 . Then φ u(p), v(p) = 0 for
all p ∈ O. It follows that
Ã
!Ã ¡
¢! Ã !
0
ux (p) vx (p)
φu T (p)
¡
¢ =
0
uy (p) vy (p)
φv T (p)
We must show that
∂(u,v)
∂(x,y) (p)
for all p ∈ O .
(A.8)
= 0 for all p ∈ O. Suppose for a contradic-
tion that there is a point p0 ∈ O such that
∂(u,v)
∂(x,y) (p0 )
6= 0. By the inverse
function theorem, we can find a neighborhood U of p0 , with U ⊆ O, and
a neighborhood V of T (p0 ) such that
(i)
∂(u, v)
(p) 6= 0
∂(x, y)
for all p ∈ U ,
¯
(ii) T ¯U : U → V is a bijective C1 -mapping whose inverse is also a
C1 -mapping.
It follows from (A.8) that
φu (e
p) = φv (e
p) = 0
e ∈V.
for all p
Hence φ ≡ 0 on a neighborhood V1 of T (p0 ) such that V1 ⊆ V . This con-
tradicts the fact that φ is not identically zero on any open set. Therefore,
∂(u, v)
(p) = 0
∂(x, y)
for all p ∈ O.
122
A. Some Results from Real Analysis
Differentiation under the Integral Sign
Theorem A.8. ([13], pp. 237–238). Suppose that A is a compact subset
of Rm , that B is an open subset of Rn , and that f : A × B → R together
with ∂f /∂tj (j = 1, . . . , n) are continuous on A × B. Then the function
φ : B → R defined by
φ(t) =
Z
f (x, t) dx
A
for all t ∈ B
is of class C1 on B and its partial derivatives are given by
Z
Z
∂
∂f
f
(x,
t)
dx
=
(x, t) dx
j
∂tj A
∂t
A
for all j ∈ {1, . . . , n} and all t ∈ B.
(A.9)
Appendix
B
Proof of Lemma 5.14
Suppose that the invariants Q, J, F are all constant. It follows from (5.18)
that β ′′′ = −ε(η − J 2 )β ′ . Note that hβ ′′ , β ′′ i = εη(η − J 2 ).
Case 1 : η − J 2 = 0. Thus η = 1 and β ′′ is a constant vector such that
hβ ′′ , β ′′ i = 0. Suppose to the contrary that β ′′ = 0; hence β ′ is a constant
vector satisfying hβ ′ , β ′ i = η = 1. It follows that there is some vector
V ∈ E31 such that β(s) = β ′ s + V , so ε = hβ, βi = s2 + 2shβ ′ , V i + hV, V i,
which is a contradiction. Hence β ′′ 6= 0. Indeed, β ′′ is a constant null
vector.
Let A, B, C be vectors in E31 such that
(i) β ′′ = A,
(ii) β ′ = As + B,
(iii) β =
and
A 2
s + Bs + C.
2
Thus hA, Ai = 0. But hβ ′ , β ′ i = 1, so 2hA, Bis + hB, Bi = 1. This implies
that hA, Bi = 0 and hB, Bi = 1. Since hβ, βi = ε, we get (hA, Ci + 1)s2 +
2hB, Cis + hC, Ci = ε. Hence hA, Ci = −1, hB, Ci = 0 and hC, Ci = ε.
Without loss of generality, we can assume that J = 1, B = (0, 0, 1)
and C = ((1 − ε)/2, (1 + ε)/2, 0). Let A = (a1 , a2 , 0) for some a1 , a2 ∈ R.
123
124
B. Proof of Lemma 5.14
Since (ABC) = J = 1 and hA, Ci = −1, we have a1 (1 + ε) − a2 (1 − ε) =
−2 = −a1 (1 − ε) + a2 (1 + ε), so a1 = a2 = −ε. Consequently,


−εs2 + 1 − ε

1
β(s) = 
−εs2 + 1 + ε

.
2
2s
(B.1)
Actually, β is the orbit of a point under a one-parameter group of Lorentzian rotations around a null axis span by (1, 1, 0). In detail:
• For ε = −1, β is precisely the orbit of the point (1, 0, 0) under the
one-parameter group of Lorentzian rotations of the 3rd type given
in Proposition 4.4, i.e.,

2
1 + s2
 2
s
β(s) = 
 2
s
2
− s2
1−
s2
2
−s
 
1
 
 
s
 0 .
0
1
s
• For ε = 1, β is the orbit of the point (0, 1, 0) under the one-
parameter group of Lorentzian rotations of the 3rd type given in
Proposition 4.4, but the parameter “s” is replaced by “−s,” i.e.,

 
2
2
1 + s2
− s2
−s
0
 2
 
2
s
 
β(s) = 
1 − s2 −s
 2
 1 .
0
−s
s
1
Since β × β ′′ = Jβ ′ = β ′ and since α′ = εF β − εQβ × β ′ , it follows that
α′′ = εF β ′ − εQβ × β ′′ = ε(F − Q)β ′ ,
α′′′ = ε(F − Q)β ′′
and α(4) = 0.
Suppose first that α′′′ = 0. Thus F − Q = 0 and α′′ = 0. This implies
that α′ = εF (β − β × β ′ ) is a constant null vector and α is a straight line.
Note that β − β × β ′ is constant; if we evaluate it at s = 0, we find that
β − β × β ′ = (1, 1, 0); hence α′ = εF (1, 1, 0). Therefore, the surface M
B. Proof of Lemma 5.14
125
is a helicoidal ruled surface which is the orbit of a straight line parallel
to the x1 -axis or the x2 -axis under a one-parameter group of Lorentzian
rotations around a null axis.
From now on suppose that α′′′ 6= 0. Then F − Q 6= 0 and α′′′ is a null
vector. Since α′′ = ε(F − Q)β ′ ,
α′ = ε(F − Q)β + V
for some V ∈ E31 .
But α′ = εF β − εQβ × β ′ , so

−εs2 − 1 − ε



1
2 − 1 + ε
ε(F − Q)β + V = εF β − εQ 
−εs


2
2s
= ε(F − Q)β + (εQ, εQ, 0).
Hence V = (εQ, εQ, 0), so
α′ = ε(F − Q)β + (εQ, εQ, 0).
We can get rid of the constants of integration by translation; we infer
that

3

 s3
α = ε F −Q
2 −ε 3
Subcase 1.1 : ε = −1. We have

α=
Note that
s3
3
Q−F  s3
2 3



εQs
 



+ εs + s
 + εQs .
s2
0
−ε s3 − εs + s

+s

− s
−
s2
Q+F
2
 
s
 
s .
 
0
126
B. Proof of Lemma 5.14
(i) the mapping
 

2
x
1 + s2
 

y  7→  s2
 
 2
z
s
2
− s2
1−
s2
2
−s
 
x
 
 
s
 y  −
1
z
 
s
 
s
 
0
s
Q+F
2
is a Lorentzian rotation around the null axis {(x, x, (Q+F )/2)
R} and that
: x∈
(ii) the mappings
 

2
x
1 + s2
 

y  7→  s2
 
 2
z
s
 
x
 
 
s
 y  −
1
z
2
− s2
1−
s
s2
2
−s
Q+F
2
 
s
 
s +
 
0

s3
3
Q−F  s3
2 3

+s

− s

2
s
form a nontrivial one-parameter group of cubic screw motions around
a null axis mentioned in (i) above.
Consequently, α is the orbit of the origin under the one-parameter group
of cubic screw motions mentioned in (ii), and the surface M with parametrization
x(s, t) = − Q+F
2
 
s
 
s +
 
0

s3
3
Q−F  s3
2 3
 
2
1 + s2
+s
  2
 s
− s
+ 2
s2
s
2
− s2
1−
s2
2
−s
 
t
 
 
s
 0
0
1
s
is the orbit of the x1 -axis under the above one-parameter group of cubic
screw motions.
Subcase 1.2 : ε = 1. As in Subcase 1.1, we can show that M is the orbit
of the x2 -axis under a one-parameter group of cubic screw motions.
Case 2 : η − J 2 6= 0. Since hβ ′ × β ′′ , β ′ × β ′′ i =
6 0, it follows that β ′
and β ′′ span a nondegenerate plane P . This plane is constant because
(β ′ × β ′′ )′ = β ′ × β ′′′ = β ′ × (−ε(η − J 2 )β ′ ) = 0.
B. Proof of Lemma 5.14
127
Since β ′ × β ′′ is constant and since
hβ − β(0), β ′ × β ′′ i′ = hβ ′ , β ′ × β ′′ i = 0 ,
we deduce that hβ − β(0), β ′ × β ′′ i = 0. This implies that β is a planar
curve. Let σ denote the sign of η − J 2 ; then the sign of the normal of P
is sgn(hβ ′ × β ′′ , β ′ × β ′′ i) = sgn(−ε(η − J 2 )) = −εσ. In the Frenet frame
e1 = β ′ , e2 = √
1
β ′′
|η−J 2 |
with signs η, ησε, we have
p
e′1 = β ′′ = |η − J 2 | e2 and
p
1
e′2 = p
β ′′′ = −εσ |η − J 2 | e1 .
|η − J 2 |
p
Hence β is a planar curve of constant curvature ησε |η − J 2 |. We will
show later that β is a circle or a rectangular hyperbola. In more detail:
η
σ
ε
β
−1 −1
1
rectangular hyperbola
1
circle
1
1
1
1 −1
1 −1
1
1 −1 −1
rectangular hyperbola
rectangular hyperbola
circle
The curve α is determined by α′′ = ε(F − QJ)β ′ . Let V ∈ E31 r {0}
be such that
α′ = ε(F − QJ) β + V.
(B.2)
Note that 0 = hV, β ′ i = hV, β ′′ i since hα′ , β ′ i = 0. Thus V is parallel to
β ′ × β ′′ .
It is easy to verify that M is a helicoidal ruled surface under a Lorentz-
ian screw motion around the axis in the direction of V . For instance, suppose that V is spacelike and (η, σ, ε) = (−1, −1, 1). Up to a Lorentzian
motion, we can assume that V = (0, 0, D) for some nonzero constant D.
Since hV, β ′ i = 0 and hβ ′ , β ′ i = −1, we may assume without loss of gen-
erality that β ′ (s) = (cosh ϕ(s), sinh ϕ(s), 0) for some smooth real-valued
128
B. Proof of Lemma 5.14
function ϕ = ϕ(s). Thus β ′′ (s) = (ϕ′ (s) sinh ϕ(s), ϕ′ (s) cosh ϕ(s), 0). In
this case, hβ ′′ , β ′′ i = 1 + J 2 , so
¡
¢2
ϕ′ (s) = 1 + J 2 .
In the moving frame e1 = β ′ , e2 =
√ 1
1+J 2
(B.3)
β ′′ , e3 = (0, 0, 1) with signs
−1, 1, 1, we have
1
β ′′ + hβ, e3 i e3
2
1
+
J
µ ′
¶
ϕ (s)
ϕ′ (s)
=
sinh ϕ(s),
cosh ϕ(s), Z ,
1 + J2
1 + J2
β=
where Z = hβ, e3 i. Since hβ, βi = 1, it follows that
1−
1
1+J 2
=
J2
.
1+J 2
Z2
= 1−
¡
¢2
ϕ′ (s)
(1+J 2 )2
=
Clearly, Z must be constant. Differentiating both sides
of (B.3), we get 2ϕ′ (s)ϕ′′ (s) = 0. If ϕ′ ≡ 0 on some open interval, then β
is a constant vector field on the same interval, thus contradicting the fact
that M is nondevelopable. We infer that ϕ′′ ≡ 0. Hence ϕ(s) = As + B
for some constants A, B with A 6= 0. By changing the parameter s, we
can assume further that A > 0 and B = 0. From (B.3), we obtain that
√
A = 1 + J 2 . Therefore,
β=
µ
¶
p
p
1
1
±|J|
2
2
√
sinh( 1 + J s), √
.
cosh( 1 + J s), √
1 + J2
1 + J2
1 + J2
This shows that β is a rectangular hyperbola on the plane x3 =
±|J|
√
.
1+J 2
It follows from (B.2) that
p
p
¢
F − QJ ¡
sinh( 1 + J 2 s), cosh( 1 + J 2 s), ±|J| + (0, 0, D).
α′ = √
2
1+J
After translation, we have
α=
p
p
p
¢
F − QJ ¡
2 s), sinh( 1 + J 2 s), ±|J| 1 + J 2 s
1
+
J
cosh(
2
1+J
+ (0, 0, Ds).
B. Proof of Lemma 5.14
129
Thus M is parametrized by
x(s, t) = α(s) + tβ(s)



√
√
F −QJ
cosh( 1 + J 2 s) sinh( 1 + J 2 s) 0
2

  1+J 
√
√

2 s) cosh( 1 + J 2 s) 0  √ t
=
1
+
J
sinh(

  1+J 2 
±|J| t
√
0
0
1
1+J 2


0


.
0
+
³
´ 
−QJ)
√
s
D ± |J|(F
1+J 2
As a result, M is a helicoidal ruled surface which is the orbit of the line
½³
¾
F − QJ
±|J| t ´
t
ℓ=
: t∈R
,√
,√
1 + J2
1 + J2
1 + J2
under a Lorentzian screw motion of the 2nd type in Proposition 4.4.
This completes the proof.
¤
Appendix
C
Ruled Surfaces in E31
Figure C.1: Helicoidal ruled surface x(s, t) = (s + t, cos s − t sin s, sin s +
t cos s)
131
132
C. Ruled Surfaces in E31
Figure C.2: Helicoid of the 1st kind x(s, t) = (as, t cos s, t sin s)
C. Ruled Surfaces in E31
133
x
z
z
y
y
x
(b) Bottom view (a = 1)
(a) Top view (a = 1)
y
z
x
y
x
(c) Z-axis view (a = 1)
(d) Front view (a = 1)
Figure C.3: Helicoid of the 2nd kind x = y tanh(z/a)
C. Ruled Surfaces in E31
134
y
z
z
x
x
y
(b) Bottom view (a = 1)
(a) Top view (a = 1)
y
z
x
x
(c) Z-axis view (a = 1)
y
(d) Front view (a = 1)
Figure C.4: Helicoid of the 3rd kind y = x tanh(z/a)
C. Ruled Surfaces in E31
135
z
z
x
x
y
y
(b) Bottom view (a = 0.5)
(a) Top view (a = 0.5)
y
z
x
y
x
(c) Z-axis view (a = 0.5)
(d) Back view (a = 0.5)
Figure C.5: Conjugate of Enneper’s surface of the 2nd kind or Cayley–Lie
¡
¢
minimal ruled surface of third degree 6a2 (x + y) = (x − y) 6az − (x − y)2
C. Ruled Surfaces in E31
136
z
z
x
y
y
x
(a) Top view a = 3, b = 0.2
(b) Bottom view a = 3, b = 0.2
y
x
z
y
x
(c) Z-axis view a = 3, b =
(d) Front view a = 3, b = 0.2
0.2
Figure C.6: Nomizu–Sasaki surface z =
x2 − y 2
+ b ln |x − y|
2a
C. Ruled Surfaces in E31
137
x
z
z
c
c
y
x
y
(b) Bottom view
(a) Top view
x
z
c
y
x
c
y
(c) Z-axis view
(d) Back view
Figure C.7: Conoid of the 1st kind x(s, t) = (exp(s), t cos s, t sin s) with
the curve c(s) = (exp(s), cos s, sin s)
C. Ruled Surfaces in E31
138
z
y
z
c
x
x
c
y
(a) Top view
(b) Bottom view
y
c
z
←c
x
x
(c) Y-axis view
(d) Z-axis view
Figure C.8: Conoid of the 2nd kind x(s, t) = (t sinh s, t cosh s, s3 +s) with
the curve c(s) = (2 sinh s, 2 cosh s, s3 + s)
C. Ruled Surfaces in E31
139
z
x
z
x
y
c
y
c
(a) Top view
(b) Bottom view
y
z
c
y
(c) X-axis view
←c
x
(d) Z-axis view
Figure C.9: Conoid of the 3rd kind x(s, t) = (t cosh s, t sinh s, s3 + s) with
the curve c(s) = (3 cosh s, 3 sinh s, s3 + s)
Appendix
D
Julius Weingarten
Julius Weingarten was born in Germany, but his family were Polish and
had emigrated to Germany. He certainly did not come from an academic
family for his father was a weaver, and the family were not well off which
would have a serious effect on the whole of Weingarten’s career.
Figure D.1: Julius Weingarten
Weingarten attended the Municipal Trade School in Berlin. He completed his studies there in 1852 and, in the same year, he entered the Uni141
142
D. Julius Weingarten
versity of Berlin to embark on a course of study which involved mainly
mathematics and physics. At the University of Berlin Weingarten attended lectures on potential theory given by Dirichlet. These lectures
were particularly inspiring and, although this would not be Weingarten’s
main area of research, he continued to work, from time to time, on problems related to this topic throughout his career. He also studied chemistry
at the Berlin Gewerbeinstitut (the Institute for Crafts) during these years.
Coming from a poor family Weingarten did not have the financial
support to allow him to complete his doctorate at Berlin without earning
his living so, in 1858, he began teaching at a school in Berlin. Despite
having to work as a teacher at various schools while he undertook research,
his work on the theory of surfaces progressed remarkably well. In fact the
work was of such quality that Weingarten received a prize for work on
the lines of curvature of a surface in 1857.
In 1864 he received a doctorate from the University of Halle for the
same work which had won him the prize from the University of Berlin,
but he had been far from idle over the years for he had published other
important work on the theory of surfaces. The theory of surfaces was the
most important topic in differential geometry and
... one of its main problems was that of stating all the surfaces
isometric to a given surface. The only class of such surfaces
known before Weingarten consisted of the developable surfaces
isometric to the plane.
In 1863 Weingarten was able to make a major step forward in the topic
when he gave a class of surfaces isometric to a given surface of revolution.
Surfaces of constant mean curvature or constant Gaussian curvature are
now called the Weingarten surfaces.
Now having produced work of outstanding quality, while one must remember he was teaching in schools, it would be reasonable to expect that
Weingarten would find a good academic post. However, this was not easy
D. Julius Weingarten
143
at that time except for those who had the necessary funds to allow them
the luxury of starting an academic career with little income. Weingarten
had to take the option which would provide him with an income so he
accepted a rather unsatisfactory position at the Bauakademie in Berlin.
Weingarten was promoted to professor at the Bauakademie in 1871
but left the rather unsatisfactory post to take on what was another rather
unsatisfactory position at the Technische Hochschule in Berlin. By 1902,
at the age of 66, his health began to fail and for that reason he moved to
Friburg im Breisgau where he was appointed as an honorary professor. He
taught there until 1908 in what was in many ways the most satisfactory
of his teaching positions.
Weingarten’s work on the infinitesimal deformation of surfaces, undertaken around 1886, was praised by Darboux who included it in his
four volume treatise on the theory of surfaces. In fact Darboux said that
Weingarten’s work was worthy of Gauss, a compliment indeed. The interest which Darboux showed in his work, encouraged Weingarten to push
his results further and he wrote a long paper which won the Grand Prix
of the Académie des Sciences in Paris in 1894. The work was published in
Acta mathematica in 1897 and was another major step forward in solving
the problems on which Weingarten had worked all his life. In this work he
reduced the problem of finding all surfaces isometric to a given surface to
the problem of determining all solutions to a partial differential equation
of the Monge-Ampère type.
Darboux was not the only leading mathematician in Weingarten’s
time who was also interested in the theory of surfaces. Another was
Bianchi and a major correspondence grew up between Weingarten and
Bianchi. In fact there is a 304 page book containing all Bianchi’s correspondence, the most extensive correspondence of all is the one with
Weingarten.
Article by: J. J. O’Connor and E. F. Robertson
Bibliography
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[3] D. E. Blair and Th. Koufogiorgos, Ruled surfaces with vanishing
second Gaussian curvature, Monatsh. Math., 113 (1992), 177–181.
[4] H. Brauner, Differentialgeometrie, Vieweg, Wiesbaden, 1981.
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Tokyo, 1978.
[6] M. P. do Carmo, Differential Geometry of Curves and Surfaces,
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, Ruled Surfaces of Weingarten Type in Minkowski 3-space,
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, Ruled W-surfaces, Arch. Math. (Basel), 62 (1994), 475–
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, Ruled surfaces with pointwise 1-type Gauss map, J. Geom.
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Index of Symbols
Note.
The entries in this Index of Symbols are grouped in alphabetical
order according to the appearance of the first letter in the symbols or in
the LATEX commands.
AvII . . . . . . . . . . . . . . . . . . . . . . . . . . 70
ei . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
(ABC) . . . . . . . . . . . . . . . . . . . . . . . . 2
En . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
{A, B}-W-surface . . . . . . . . . . . . 85
En1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Aff n (k) . . . . . . . . . . . . . . . . . . . . . . 39
Ep . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Ak (V ) . . . . . . . . . . . . . . . . . . . . . . . 17
exp . . . . . . . . . . . . . . . . . . . . . . . . . . 48
r
Aij11,...,i
,...,js . . . . . . . . . . . . . . . . . . . . . . 23
AT
E31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
. . . . . . . . . . . . . . . . . . . . . . . . . . .41
F(M ) . . . . . . . . . . . . . . . . . . . . . . . . 15
C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
b k . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Γ
ij
Cji . . . . . . . . . . . . . . . . . . . . . . . . . . . .33
γ ′ (t) . . . . . . . . . . . . . . . . . . . . . . . . . 47
∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
b . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
∆
GLn (k) . . . . . . . . . . . . . . . . . . . . . . 36
grad . . . . . . . . . . . . . . . . . . . . . . . . . 15
δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
H . . . . . . . . . . . . . . . . . . . . . . . . . . . .66
δji . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
HI . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
div. . . . . . . . . . . . . . . . . . . . . . . . . . .16
c . . . . . . . . . . . . . . . . . . . . . . . . . . 75
div
HII . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Hn (r), Hn+ (r), Hn− (r) . . . . . . . . . 44
dxi . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Hom . . . . . . . . . . . . . . . . . . . . . . . . . 20
E . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Hom(E, F ). . . . . . . . . . . . . . . . . . .25
149
150
In . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
iX . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
K . . . . . . . . . . . . . . . . . . . . . . . . . . . .61
k . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Index of Symbols
⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
∂i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
∂M+ , ∂M− , ∂M0 . . . . . . . . . . . . . 18
π : E → M . . . . . . . . . . . . . . . . . . .24
KII . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Ψ|i . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
h· , ·i . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Qn . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
LorAff n (R) . . . . . . . . . . . . . . . . . . 54
LorAff n (+, ↑) . . . . . . . . . . . . . . . . 54
Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
⋊ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Lor(n) . . . . . . . . . . . . . . . . . . . . . . . 45
sgn . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Lorn (+, ↑) . . . . . . . . . . . . . . . . . . . 46
SLn (k) . . . . . . . . . . . . . . . . . . . . . . . 36
⋉ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
M11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
SO(n) . . . . . . . . . . . . . . . . . . . . . . . 42
SO1 (n) . . . . . . . . . . . . . . . . . . . . . . 44
M01 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Tkij . . . . . . . . . . . . . . . . . . . . . . . . . . 69
M0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Trs (M ) . . . . . . . . . . . . . . . . . . . . . . . 21
Mn (k) . . . . . . . . . . . . . . . . . . . . . . . 35
Trs (V ). . . . . . . . . . . . . . . . . . . . . . . .20
Mm,n (k) . . . . . . . . . . . . . . . . . . . . . 35
Tp∗ M . . . . . . . . . . . . . . . . . . . . . . . . .24
M ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Transn (k) . . . . . . . . . . . . . . . . . . . . 40
∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
b . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
∇
⊳ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
∇i γ|j . . . . . . . . . . . . . . . . . . . . . . . . 30
∇∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
NMn (k) (A; r) . . . . . . . . . . . . . . . . . 38
O(·) . . . . . . . . . . . . . . . . . . . . . . . . . 71
O(n) . . . . . . . . . . . . . . . . . . . . . . . . . 41
O1 (n) . . . . . . . . . . . . . . . . . . . . . . . . 43
⊕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
O+ (n), O− (n) . . . . . . . . . . . . . . . . 42
k · k . . . . . . . . . . . . . . . . . . . . . . . . . . 36
| · |............................3
V ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Index
B
curvature, 4
B -scroll, 10
Gaussian, 61
base, 24
second, 68
Brioschi’s formula, 64
mean, 66
bundle
second, 78
cotangent, 24
principal, 9
tangent, 24
curve, 47
vector, 24
base, 6
differentiable, 47
C
director, 6
Cayley–Lie minimal ruled surface
smooth, 47
of third degree, 87
Conjugate of Enneper’s surface
D
of the 2nd kind, 87
derivative, 47
conoid
difference tensor, 69
Lorentzian, 88
divergence, 16
of the 1st kind, 88
divergence theorem, 18
of the 2nd kind, 89
E
of the 3rd kind, 89
equations
contraction, 32
Gauss, 13
over i, j, 33
covariant derivative, 5
Weingarten, 14
151
152
Index
exponential, see matrix exponential
special linear, 36
translation, 40
unimodular, 36
F
fiber, 24
H
fiber metric, 28
helicoid
Frenet formulas, 3
Lorentzian, 86
Frenet frame, 3
of the 1st kind, 86
function
of the 2nd kind, 86
signum, 71
of the 3rd kind, 87
functional relation, 121
of class
Cr ,
homomorphism, 25
121
I
functionally dependent, 120
interior contraction, 17
G
gradient, 15
group
interior multiplication, 17
isometry, 42
in M , 42
affine, 39
isomorphism, 25
Lorentzian, 54
K
proper orthochronous Lorentzian, 54
Kronecker delta symbol, 11
general linear, 36
Lorentzian, 43
proper, 44
proper orthochronous, 46
matrix, 39
L
Lagrange identity, 2
Laplacian, 17
local trivialization, 24
M
one-parameter, 47
trivial, 49
matrix
orthogonal, 41
orthogonal, 41
special, 42
unimodular, 36
rotation
linear Lorentzian, 43
matrix exponential, 48
metric tensor
Index
153
R
Lorentzian, 2
Riemannian, 1
Ricci lemma, 31
rotation
motion
cubic screw, 58
axis of, 45
Euclidean screw, 59
Lorentzian
Lorentzian, 54
around an axis, 45
Lorentzian screw, 56
linear, 43
ruling, 6
N
norm
of a matrix, 36
of a vector, 3
nullcone, 43
S
section, 25
shape operator, 9
space
Minkowski, 42
total, 24
P
parametrization, 5
striction line, 7
subgroup
standard, 8
patch, 4
in M , 4
coordinate, 4
matrix, 39
sum
direct, 19
surface
proper, 4
developable, ix
regular, 4
helicoidal, 85
product
Nomizu–Sasaki, 87
direct, 19
nondevelopable, ix
scalar
ruled, 6
Lorentzian, 42
semidirect, 40
conoidal, 6
cylindrical, 6
projection, 24
helicoidal, 85
pseudohyperbolic space, 6
noncylindrical, 6
pseudosphere, 6
orthogonal, 8
154
Index
simple, 5
outward-pointing, 17
Weingarten, 85
spacelike, 3
T
tensor, 20
components of, 23
contravariant, 23
timelike, 3
vector field
director, 6
W
covariant, 23
W-surface, see surface, Weingarten
mixed, 23
Weingarten, Julius, 143
tensor field, 21
parallel, 31
tensor product
of modules, 20
of tensor fields, 21
torsion, 4
transformation
affine, 40
Lorentzian
linear, 43
transpose, 41
II-area, 70
II-flat, 79
II-minimal, 79
V
variation
first, 78
normal, 70
vector
inward-pointing, 17
lightlike, see vector, null
null, 3