ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS
IN EUCLIDEAN SPACES
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
Abstract. We introduce the concepts of rounding and flattening of a smooth map g
of m-dimensional manifold M to the eucledean space Rn with m < n, as those points in
M such that g(M ) has contact of type Σ(m,...,m) with a hypersphere or a hyperplane of
Rn respectively. This includes several known special points like vertices or flattenings of
a curve in Rn , umbilics of a surface in R3 , or inflections of a surface in R4 .
1. Introduction
In [24], Porteous studied the Thom-Boardman singularities Σi1 ,...,ik of the distancesquared unfolding associated with a smooth embedding g : M → Rn from a smooth
m-dimensional manifold M into Rn , which is defined by
D : Rn × M −→ Rn × R,
(x, t) 7−→ (x, dx (t)),
where
1
dx (t) = kx − g(t)k2 .
2
He discovered that for small values of m and n, these singularities give a new point of
view of some well-known facts of classical differential geometry. This work was continued by Montaldi [19], who introduced the notion of contact between submanifolds and
deduced that the singularity types of D characterize the contact types of g(M ) with the
hyperspheres of Rn . Moreover, he also shows that for a residual set in C ∞ (M, Rn ) with
the Whitney C ∞ topology, the associated unfolding D is generic in the Thom-Boardman
sense (that is, its k-jet extension is transverse to the strata Σi1 ,...,ik ).
For instance, suppose that g : I → R2 defines a regular smooth curve in the plane.
Then, a point of type Σ1,1,1 of D is a pair (x, t) such that t ∈ I is a vertex of the curve
(that is, a point where κ0 = 0) and x ∈ R2 is the center of curvature at that point. If
now g : M 2 → R3 is a smooth immersion of a surface in R3 , a point of type Σ2,2 of D is a
pair (x, t), where t ∈ M is an umbilic (that is, the two principal curvatures coincide) and
x ∈ R3 is the center of principal curvature.
We can do a similar analysis if we look at the height unfolding, given by
H : S n−1 × M −→ S n−1 × R,
(v, t) 7−→ (v, hv (t)),
where
hv (t) = hv, g(t)i.
1991 Mathematics Subject Classification. Primary 53A07; Secondary 58K05, 53A05.
Key words and phrases. Distance squared function, height function.
The second author has been partially supported by DGICYT Grant PB96– 0785.
1
2
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
In this case, the singularity types of H determine the contact type of g(M ) with the
hyperplanes of Rn and it is also true that for a residual set in C ∞ (M, Rn ) the unfolding
H is generic in the Thom-Boardman sense. For instance, if g : I → R3 is a regular curve
with a well defined Frenet frame, then the Σ1,1,1 points of H are the pairs (v, t) where
t ∈ I is a flattening (that is, the torsion is zero at t) and v ∈ S 1 is the binormal vector. If
g : M 2 → R4 is a smooth immersion of a surface in R4 , a Σ2,2 point of H is a pair (v, t),
where t ∈ M is an inflection (that is, the two second fundamental forms are proportional)
and v ∈ S 3 is the binormal vector in the sense of [8].
In the last years, these concepts have been of great interest for people working on
differential geometry and singularities. On one hand, we have the classical Four Vertex
Theorem, which states that any closed simple regular plane curve has at least four vertices,
and its generalization to higher dimensions (see for instance, [1]). On the other hand, the
Caratheodory Conjecture says that every smooth immersion of the sphere S 2 into R3 has
at least two umbilics (See Remark 4.7). Moreover, it has been shown recently (cf. [8])
that any convex generic immersion of the sphere S 2 into R4 has at least four inflections.
In Section 2, we introduce the concept of k-rounding (respectively k-flattening) as a
point t0 ∈ M which corresponds to a point of type Σm(k) of D (resp. H). Here m(k)
denotes the Boardman symbol (m, . . . , m), with m repeated k times. Note that these
definitions have sense even when the map g : M → Rn has singularities. Then, for each
m and k we can choose n, the dimension of the ambient space, such that the k-roundings
or the k-flattenings of a generic smooth map g : M m → Rn appear as isolated points (see
Table 1).
Associated with a k-rounding or k-flattening we consider a local algebra of contact,
which is obtained by eliminating the parameter x or v in the local algebra of contact of D
or H, respectively, with the stratum Σm(k) . It follows that this local algebra of contact is
a local geometric invariant (that is, it is invariant under change of coordinates in M and
isometries in Rn ). Moreover, from this algebra, we can also obtain numerical invariants
like its multiplicity or its index (in the orientable case). The index of a k-rounding or
k-flattening is nothing but the oriented intersection number of the k-jet extension of the
dx or hv with the stratum Σm(k) . At the end of Section 2, we show that roundings and
flattenings are related through the stereographic projection. We see that a point t0 ∈ M
is a rounding of g : M → Rn if and only if it is a flattening of ξ ◦ g : M → Rn+1 , where
ξ : Rn → S n ,→ Rn+1 denotes the inverse of the stereographic projection. Moreover, the
index is also preserved.
In Section 3, we study curves in Rn and give an explicit formula to compute roundings
(that is, vertices) which seems to be new.
Sections 4 and 5 are dedicated to surfaces in R3 . In the case of an isolated umbilic
of a surface in R3 , we show that the index we have defined is essentially the index of
the foliation defined by curvature lines. We also provide formulas for these indices which
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
3
work even for isolated singular points of the surface. In Section 5, we introduce complex
coordinates in regular surfaces, in order to compute indices of umbilic points by looking
at the coefficients in a Monge normal form.
Finally, in Section 6 we study surfaces in R4 . Then, we see that for an isolated inflection
of a surface in R4 , the index is essentially the index of the foliation defined by asymptotic
lines.
The authors would like to thank M.C. Romero-Fuster, A. Montesinos-Amilibia and S.
Izumiya for helpful comments. The authors also would like to thank to the referee for
many comments on the earlier version, and informing the authors controversy about proof
of the Carathéodory conjecture.
2. Roundings and flattenings
2.1. Definitions of roundings and flattenings. Let M be a smooth m-manifold and
let Sk denote the Thom-Boardmabn submanifold Σm(k) in the jet space J r (M, R), k ≤ r,
which is the subset of the jet space with all partial derivatives of degree ≤ k equal to zero.
Definition 2.1. Let g : M → Rn be a smooth map from a smooth m-dimensional
manifold M into Rn , with m < n. Let k, r be the positive integers with k ≤ r.
We say that p ∈ M is a k-flattening of g if there is v ∈ S n−1 such that j r hv (p) ∈ Sk .
We say that p ∈ M is a k-rounding of g if there is x ∈ Rn such that j r dx (p) ∈ Sk .
We say that p ∈ M is a non-flat k-rounding of g if it is a k-rounding but not a kflattening.
When k is clear from the context we will just say that p is a flattening or a rounding.
Consider Thom-Boardman classes of the unfoldings D(x, t) = (x, dx (t)) and H(v, t) =
(v, hv (t)). Since they have corank one, we have that (v, t) ∈ Σm(k) (H) (resp. (x, t) ∈
Σm(k) (D)) if and only if j r hv (p) ∈ Sk (resp. j r dx (p) ∈ Sk ).
Assume that g : M → Rn is an immersion. It follows from the definition that if
p ∈ M is a k-flattening of g and v ∈ S n−1 such that (v, p) ∈ Σm(k) (H), then g(M ) has a
contact of type Σm(k) at g(p) with the hyperplane of Rn orthogonal to v through the point
g(p). Analogously, if p is a non-flat k-rounding and x ∈ Rn such that (x, p) ∈ Σm(k) (D),
then g(M ) has a contact of type Σm(k) at g(p) with the hypersphere centered at x and
radius R = kx − g(p)k. Note that our definitions make sense even in the case that g has
singularities.
We see now that it is possible to eliminate the parameter v or x in the definition and
obtain a condition just in terms of p. Let us denote it by ν = ν(m(k) ) the cardinality of
the set {α = (α1 , . . . , αm ) : 1 ≤ |α| ≤ k}, so that ν gives exactly the number of local
equations which define Σm(k) . This number can be easily computed by the combinatorial
formula
µ
¶
m+k
ν=
− 1.
m
4
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
Moreover, we will fix some order in this set (for instance, the degree lexicographical order
associated with the monomials tα ).
Definition 2.2. Let g : M → Rn be a smooth map. Given a local coordinate system
φ : U ⊂ M → Rm , we denote g̃ = g ◦ φ−1 and φ(p) = t, for each p ∈ U . We define the
matrices:
¯
µ |α|
¶
¶
µ |α|
¯
∂ g̃i
∂ g̃i
k
k
¯
F (g̃; p) =
(t)
(t) ¯ ϕα (t)
, and R (g̃; p) =
,
1≤i≤n
1≤i≤n
∂tα
∂tα
1≤|α|≤k
1≤|α|≤k
where the functions ϕα are defined by induction on |α| in the following way: If |α| = 1,
then ϕα = 0. If |α| > 1, there are β and i ∈ {1, . . . , m} such that α = β+(0, . . . , 1i) , . . . , 0);
then,
D ∂g̃ ∂ |β| g̃ E ∂ϕ
β
ϕα =
, β +
.
∂ti ∂t
∂ti
The matrices F k (g̃; p) and Rk (g̃; p) have size ν × n and ν × (n + 1), respectively. Again,
when k is clear from the context we shall denote the matrices just by F (g̃; p) and R(g̃; p).
Lemma 2.3. Let g : M → Rn be a smooth map and let p ∈ M . Then,
(1) p is a k-flattening of g if and only if rank F k (g̃; p) < n;
(2) p is a k-rounding of g if and only if rank Rk (g̃; p) < n + 1.
Proof. Since hv ◦ φ−1 (t) = hv, g̃(t)i, we have
D ∂ |α| g̃ E
∂ |α| (hv ◦ φ−1 )
(t)
=
v, α (t) ,
∂tα
∂t
so that p is a k-flattening if and only if there is v ∈ S n−1 such that
D ∂ |α| g̃ E
v, α (t) = 0, ∀α : 1 ≤ |α| ≤ k.
∂t
This is a homogeneous linear system in the variables v1 , . . . , vn , whose coefficient matrix
is F (g̃; p). It follows that there is a solution v ∈ S n−1 if and only if F k (g̃; p) has rank < n.
For the distance-squared functions, we have that dx ◦ φ−1 (t) = 21 kx − g̃(t)k2 and hence,
D
∂ |α| (dx ◦ φ−1 )
∂ |α| g̃ E
(t)
=
−
x
−
g̃(t),
(t) + ϕα (t).
∂tα
∂tα
Now, p is a non-flat k-rounding if and only if it is not a k-flattening and there is x ∈ Rn
such that
D
∂ |α| g̃ E
x − g̃(t), α (t) = ϕα (t), ∀α : 1 ≤ |α| ≤ k.
∂t
In this case, we have a non-homogeneous linear system in the variables Xi = xi − gi (t),
i = 1, . . . , n, with coefficient matrix F k (g̃; p) and with independent terms (ϕα (t)). It
follows that the system has a solution if and only if rank F k (g̃; p) = rank Rk (g̃; p) = n.
Finally, note that
rank F k (g̃; p) ≤ rank Rk (g̃; p) ≤ rank F k (g̃; p) + 1,
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
which completes the proof.
5
¤
It follows from the proof of the above lemma that if p ∈ M is a non-flat k-rounding of
g : M → Rn , then there is a unique x ∈ Rn such that j k dx (p) ∈ Sk (i.e., (x, p) ∈ Σm(k) (D)).
Example 2.4. In the next sections, we will see that roundings and flattenings coincide
with classical concepts in differential geometry of curves and surfaces:
(1) Let α : I → R2 be a regular plane curve. Then t ∈ I is a 2-flattening if and only
if it is an inflection (that is, the curvature κ is zero at t). The vertices of α (that
is, the points where κ0 (t) = 0) correspond to the 3-roundings of α.
(2) Suppose now that α : I → R3 is a regular space curve with non-vanishing curvature, so that it has a well defined Frenet frame. Then, a 3-flattening is a point
where the torsion τ is zero.
(3) Let g : M 2 → R3 be a regular surface in R3 . Then 2-roundings correspond to
umbilic points of g (that is, points where the two principal curvatures are equal).
(4) Finally, if g : M 2 → R4 is a regular surface in R4 , the 2-flattenings of g are known
as inflections and correspond to points where the two second fundamental forms
are linearly dependent.
Definition 2.5. Let g : U ⊂ Rm → Rn be a smooth map. Given t0 ∈ U we denote
by C ∞ (Rm , t0 ) the local algebra of smooth function germs from (Rm , t0 ) to R. Then, we
define R(g, t0 ) as the ideal in C ∞ (Rm , t0 ) generated by the germs at t0 of the (n+1)-minors
of R(g, t).
Analogously, we can define F(g, t0 ) as the ideal in C ∞ (Rm , t0 ) generated by the germs
at t0 of the n-minors of F (g, t).
The following lemma shows that although the matrices R(g, t) and F (g, t) depend on the
choice of coordinates in Rm , the corresponding ideals R(g, t0 ) and F(g, t0 ) are coordinate
independent.
Lemma 2.6. Let g : U ⊂ Rm → Rn be a smooth map and t0 ∈ U . Suppose that
φ : (Rm , t00 ) → (Rm , t0 ) is a diffeomorphism germ and denote by φ∗ : C ∞ (Rm , t0 ) →
C ∞ (Rm , t00 ) the induced isomorphism of local algebras. Then, φ∗ (R(g, t0 )) = R(g ◦ φ, t00 )
and φ∗ (F(g, t0 )) = F(g ◦ φ, t00 ).
Proof. Let us denote by ei the m-tuple (0, . . . , 1i) , . . . , 0). We define the function germs
λαβ ∈ C ∞ (Rm , t00 ), with 1 ≤ |β| ≤ |α| by induction on |α| in the following way:
λei ,ej =
∂φj
∂ti
and if |α| > 1 and |α| ≥ |β|, then
m
(2.1)
λα+ei ,β
∂λαβ X
+
λei ,ej λα,β−ej ,
=
∂ti
j=1
6
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
where we take the convention that λαβ = 0 when it is not defined. Then, by direct
application of the chain rule we obtain that for any α, with |α| = r,
¶
µ |β|
X
∂ r (g ◦ φ)
∂ g
=
λαβ
◦φ .
∂tα
∂tβ
(2.2)
1≤|β|≤r
In fact, we have
¶!
∂ |β| g
λαβ
◦φ
∂tβ
β
Ã
!
¶
X ∂λαβ ∂ |β| g
X
Xµ ∂ |β|+1 g
∂φj
=
◦φ+
λαβ
◦φ
∂ti ∂tβ
∂tβ+ej
∂tj
j
β
β
Ã
¶!
µ |β|
X ∂λαβ ∂ |β| g
X X
∂ g
=
◦φ+
◦φ
λαβ−ej λei ,ej
β
β
∂t
∂t
i ∂t
j
β
β
¶
µ
|β|
X
∂ g
λα+ei ,β
◦φ .
=
β
∂t
β
∂ r+1 (g ◦ φ)
∂
=
α+e
i
∂t
∂ti
Ã
X
µ
This implies that any n-minor d of F k (g ◦ φ, t) can be written as a linear combination
P
d=
ai φ∗ (di ), where di are n-minors of F k (g, t) and ai ∈ C ∞ (Rm , t00 ). This gives the
inclusion F(g ◦ φ, t00 ) ⊂ φ∗ (F(g, t0 )) and the opposite inclusion follows by applying the
same argument to g̃ = g ◦ φ and φ̃ = φ−1 .
To conclude that φ∗ (R(g, t0 )) = R(g ◦ φ, t00 ), we just need to prove that
X
ϕ0α =
(2.3)
λαβ (ϕβ ◦ φ) ,
1≤|β|≤r
where ϕ0α are the corresponding functions for g ◦ φ. Then, (2.3) follows from (2.2) and
(2.1) by induction on r = |α|:
ϕ0α+ei
D ∂(g ◦ φ) ∂ r (g ◦ φ) E ∂ϕ0
α
,
+
=
α
∂ti
∂t
∂ti
D ∂g
E X ∂λ
³ ∂ϕ
´
|β|
X
X
∂ g
αβ
β
=
λei ,ej λαβ
◦ φ, β ◦ φ +
(ϕβ ◦ φ) +
λei ,ej λαβ
◦φ
∂tj
∂t
∂ti
∂tj
j,β
β
j,β
¶ X
µD
E ∂ϕ
X
∂ |β| g
∂λαβ
∂g
β
◦ φ, β ◦ φ +
◦φ +
(ϕβ ◦ φ)
=
λei ,ej λαβ
∂t
∂t
∂t
∂t
j
j
i
β
j,β
=
X
j,β
λei ,ej λαβ (ϕβ+ej ◦ φ) +
X ∂λαβ
β
∂ti
(ϕβ ◦ φ)
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
=
X
λei ,ej λα,β−ej (ϕβ ◦ φ) +
j,β
=
X
X ∂λαβ
β
∂ti
7
(ϕβ ◦ φ)
λα+ei ,β (ϕβ ◦ φ).
β
¤
2.2. Multiplicities of flattenings and roundings.
Definition 2.7. Let g : M → Rn be a smooth map from a smooth m-dimensional
manifold M into Rn . Given p ∈ M , we denote by C ∞ (M, p) the local algebra of smooth
function germs from (M, p) into R. If we choose a local chart φ : (M, p) → (Rm , t0 ), we
can consider the ideals R(g ◦ φ−1 , t0 ) and F(g ◦ φ−1 , t0 ) in C ∞ (Rm , t0 ). We define the
ideals R(g, p) and F(g, p) in C ∞ (M, p) by
R(g, p) = φ∗ R(g ◦ φ−1 , t0 ),
F(g, p) = φ∗ F(g ◦ φ−1 , t0 ),
where φ∗ : C ∞ (Rm , t0 ) → C ∞ (M, p) is the induced isomorphism of local algebras. The
above lemma shows easily that this construction does not depend on the chosen chart φ.
Moreover, we also define the multiplicity of a rounding or a flattening, respectively, as
µR (g, p) = dimR C ∞ (M, p)/R(g, p),
µF (g, p) = dimR C ∞ (M, p)/F(g, p).
If p ∈ M is in fact a rounding or a flattening, we have that µR (g, p) ≥ 1 or µF (g, p) ≥ 1,
respectively. Moreover, it is also true that if µR (g, p) < ∞ or µF (g, p) < ∞, then p is an
isolated rounding or flattening, respectively.
Definition 2.8. Let g : M → Rn be a smooth map from a smooth m-dimensional
manifold M into Rn . We say that g is generic if the unfoldings D, H are generic in
the Thom-Boardman sense (that is, their k-jet extensions are transverse to the ThomBoardman strata). In particular, it follows that Σm(k) (D) and Σm(k) (H) are submanifolds
of codimension ν = ν(m(k) ) of Rn × M and S n−1 × M respectively. Thus:
(1) If ν = n + m and g is generic, the roundings of g are non-flat, appear as isolated
points and have multiplicity one.
(2) If ν = n + m − 1 and g is generic, the flattenings of g have rank n − 1, appear as
isolated points and have multiplicity one.
See [32] for the proof of the generality of the generic map, that is, any map can be
approximated by a sequence of generic maps.
In Table 1, we show the value of n = ν(m(k) ) − m, so that roundings of a map g :
M m → Rn appear generically as isolated points. By adding one to all the entries of the
table, we get n = ν(m(k) ) − m + 1, which corresponds to the dimensions where flattenings
appear generically as isolated points.
8
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
m\k
1
2
3
4
5
2 3 4
5
6
7
8
9
10
1 2
3
4
5
6
7
8
9
3 7 12 18 25 33
42
52
63
6 16 31 52 80 116 161 216 282
10 30 65 121 205 325 490 710 996
15 50 120 246 456 786 1281 1996 2997
Table 1
When the map g : M → Rn is not generic, we can take a generic deformation gλ :
M → Rn , λ ∈ (−², ²), which means that g0 = g, gλ is generic for λ 6= 0 and the map
(λ, t) 7→ gλ (t) is smooth. If t0 ∈ M is a rounding or a flattening of g, we can ask about
the number of roundings or flattenings that appear in gλ near p, for λ 6= 0 small enough.
We will denote this numbers by nR (gλ , p) or nF (gλ , p), respectively. In the following
theorem, we see that in the analytic case, such numbers are related to the corresponding
multiplicities.
Theorem 2.9. Suppose that g : M m → Rn is analytic and let gλ : M → Rn , λ ∈ (−², ²),
be an analytic generic deformation of g.
(1) If p ∈ M is a rounding of finite multiplicity and ν(m(k) ) = n+m, then nR (gλ , p) ≤
µR (g, p) and nR (gλ , p) ≡ µR (g, p) modulo 2, for λ 6= 0 small enough.
(2) If p ∈ M is a flattening of finite multiplicity and ν(m(k) ) = n + m − 1, then
nF (gλ , p) ≤ µF (g, p) and nF (gλ , p) ≡ µF (g, p) modulo 2, for λ 6= 0 small enough.
Proof. As usual, by taking coordinates we can suppose that M = U is an open subset of
Rm . Then, g : U → Rn induces a complex analytic map ĝ : Û → Cn , where Û is an open
neighbourhood of p in Cm . We will denote by OCm ,p the local ring of complex analytic
function germs from (Cm , p) to C, and R(ĝ, p) will denote the ideal in OCm ,p generated by
the maximal minors of the matrix R(ĝ, t) defined in an obvious way. Moreover, it follows
that
OCm ,p /R(ĝ, p) ∼
= C ∞ (Rm , p)/R(g, p) ⊗ C,
so that
µR (g, p) = dimR C ∞ (Rm , p)/R(g, p) = dimC OCm ,p /R(ĝ, p).
On the other hand, the deformation gλ : U → Rn also induces a complex analytic
deformation ĝλ : Û → Cn , λ ∈ V̂ , where now V̂ is an open neighbourhood of 0 in C. We
will denote by nR (ĝλ , p) the number of complex roundings of ĝλ near p. It follows that
nR (gλ , p) ≤ nR (ĝλ , p) and nR (gλ , p) ≡ nR (ĝλ , p) and thus, it is enough to prove that for
λ 6= 0 small enough,
nR (ĝλ , p) = dimC OCm ,p /R(ĝ, p).
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
9
To see this, we use classical arguments of complex analytic geometry. To simplify
notation, let us denote by A the ring OC×Cm ,(0,p) and by I the ideal in A generated by
the maximal minors of the matrix R(ĝλ , t). The complex analytic set germ of zeros of I,
V (I), is given by the pairs (λ, t) in a neighbourhood of (0, p) such that t is a rounding of
gλ . Thus, the number nR (ĝλ , p) is equal to the cardinality of π −1 (λ), where π is the finite
map germ π : V (I) → C, given by π(λ, t) = λ. By the Samuel Formula (see for instance
[22]), π −1 (λ) for λ 6= 0 small enough, is equal to the multiplicity e(hλi; A/I), where λ is
the class of λ in A/I.
Now, I is defined by the maximal minors of a matrix of size ν × (n + 1) and a result
by Hochster-Eagon [12] ensures that V (I) has dimension ≥ m + 1 − (ν − (n + 1) + 1) = 1
and if the equality holds, then A/I is a Cohen-Macaulay ring. Note that the finiteness
of π implies that the dimension of V (I) is ≤ 1. Thus, V (I) is in fact 1-dimensional and
A/I is Cohen-Macaulay.
Finally, use Theorem 17.11 of [17] to conclude that since A/I is a 1-dimensional CohenMacaulay ring and λ is a parameter system in A/I. Then
e(hλi; A/I) = dimC (A/I)/hλi = dimC OCm ,p /R(ĝ, p).
The proof of the second part of the theorem is exactly the same.
¤
2.3. Oriented intersection numbers and indices. Let M be an oriented manifold
and let S be a co-oriented submanifold of a manifold N with dim M = codim S. Assume
that f : (M, p) → (N, f (p)) is a smooth map germ with f −1 (S) = p. We denote the
oriented intersection number of f and S at p by Ip (f, S).
Some basic properties of the oriented intersection number are the following:
(1) If f is transverse to S at p, then Ip (f, S) = ±1.
(2) Assume that S is locally given near f (p) by S = g −1 (0), where g : (N, f (p)) →
(Rm , 0) is a smooth map germ with 0 as a regular value so that the co-orientation
of S in N coincides with the orientation of Rm via g. Then Ip (f, S) is equal to the
mapping degree of the map germ g ◦ f : (M, p) → (Rm , 0).
We now introduce the concept of index of a non-flat isolated rounding or an isolated
flattening of rank n − 1, which can be seen as an oriented version of multiplicity. Asuume
that M is an orientee manifold. Note that Sk is a co-oriented submanifold of J k (M, R),
whose co-orientation is determined by choosing an order in the monomials tα .
Definition 2.10. Let g : M → Rn be a smooth map from an oriented m-manifold into Rn .
Let p ∈ M be a non-flat isolated rounding of g and suppose also that ν = ν(m(k) ) = n+m.
There exists a unique x0 ∈ Rn such that (x0 , p) ∈ Σm(k) (D). Then, we define the index of
the rounding p, indR (g, p), as the oriented intersection number of the map
(Rn × M, (x0 , p)) −→ (J k (M, R), jxk0 (t)),
(x, t) 7−→ j k dx (t),
10
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
and Sk at (x0 , p). By taking local coordinates we can assume that M = U ⊂ Rm , an open
subset. Then, indR (g, p) is equal to the mapping degree of the map germ
µ |α| ¶
∂ dx
n
m
ν
θR : (R × R , (x0 , p)) −→ (R , 0),
(x, t) 7−→
.
∂tα 1≤|α|≤k
Definition 2.11. Let g : M → Rn be a smooth map from an oriented m-manifold into
Rn . Assume that the sphere S n−1 is oriented. Let p ∈ M be an isolated flattening of g
of rank n − 1 and suppose also that ν = ν(m(k) ) = n + m − 1. There is v0 ∈ S −1 such
that (v0 , p) ∈ Σm(k) (H). Then, we define the index of the flattening p, indF (g, p), as the
oriented intersection number of the map
(S n−1 × M, (v0 , p)) −→ (J k (M, R), j k hv0 (p)),
(v, t) 7−→ j k hv (t),
and Sk at (v0 , p). Again, we can take local coordinates and assume that M = U ⊂ Rm ,
an open subset. Then, indF (g, p) is equal to the mapping degree of the map germ
µ |α| ¶
∂ hv
n−1
m
ν
.
θF : (S
× R , (v0 , p)) −→ (R , 0),
(v, t) 7−→
∂tα 1≤|α|≤k
Note that we have exactly two choices for the vector, namely ±v0 . Thus, when ν =
n + m − 1 is even, indF (g, p) is well defined and does not depend on v0 . However, when
ν = n + m − 1 is odd, we can only consider | indF (g, p)|.
It follows that if g : M → Rn is generic and ν = ν(m(k) ) = n + m, then all its roundings
are non-flat, isolated and have index ±1. When g is not generic, but the rounding p ∈ M
is non-flat an isolated, we can use the Thom’s transversality theorem and take a generic
deformation gλ : M → Rn , λ ∈ (−², ²), so that the index is equal to the sum of the indices
of the generic roundings that appear in gλ near p. Moreover, we have an analogous
statement for flattenings.
Corollary 2.12. Let g : M → Rn be a smooth map from an oriented smooth m-manifold
in Rn .
(1) If p ∈ M is a rounding of finite multiplicity and ν(m(k) ) = n+m, then | indR (g, p)| ≤
µR (g, p), and indR (g, p) ≡ µR (g, p) modulo 2.
(2) If p ∈ M is a (n − 1)-rank flattening of finite multiplicity and ν(m(k) ) = n + m − 1,
then | indF (g, p)| ≤ µF (g, p), and indF (g, p) ≡ µF (g, p) modulo 2.
As another consequence, we obtain that when the rounding is non-flat or when the
flattening has rank n − 1, then Theorem 2.9 is also true even when the maps are smooth
instead of analytic.
We finish this section by showing that roundings and flattenings are related through the
stereographic projection. Let ξ : Rn → S n ,→ Rn+1 denote the inverse of the stereographic
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
11
projection, which is given by
ξ(x) =
(2x, kxk2 − 1)
.
(kxk2 + 1)
It follows from the works by Romero Fuster [26],[27] and Sedykh [29] that p ∈ M is a
rounding of a smooth map g : M → Rn if and only if it is a flattening of ξ ◦ g : M → Rn+1 .
In fact, if p ∈ M is an isolated non-flat rounding of g, then p is an isolated flattening of
ξ ◦g of rank n−1. This means that there is a unique x0 ∈ Rn such that (x0 , p) ∈ Σm(k) (D).
However, there are two possible choices for the vector v0 ∈ S n , so that (v0 , p) ∈ Σm(k) (H)
and H being the height unfolding of ξ ◦ g. We will see that there is a canonical choice of
this vector v0 in such a way that indR (g, p) = indF (ξ ◦ g, p).
Given a ∈ Rn and r > 0, we define
v(a, r) = −(2a, kak2 − r2 − 1),
u(a, r) =
v(a, r)
,
kv(a, r)k
ρ(a, r) = −
kak2 − r2 + 1
.
kv(a, r)k
Then, by direct computation we have:
(1) If r = kx0 −ak, then hξ(x0 ), u(a, r)i = ρ(a, r). This means that ξ maps the (n−1)sphere in Rn with center a and radius r into the (n − 1)-sphere in S n given by the
intersection with the hyperplane in Rn+1 whose equation is hy, u(a, r)i = ρ(a, r).
Obviously, this is also true if we take −u(a, r) and −ρ(a, r).
(2) For any x ∈ Rn ,
hξ(x), u(a, r)i − ρ(a, r) =
2
(kx − ak2 − r2 ).
kv(a, r)k(kxk2 + 1)
Thus, the choice of u(a, r) and ρ(a, r) is so that we obtain the height function by
multiplying the distance squared function by a positive function. The other choice
of u(a, r) and ρ(a, r) would give a product with a negative function.
Let p ∈ M be an isolated non-flat rounding of g and let x0 ∈ Rn be such that (x0 , p) ∈
Σm(k) (D). Then the map u(x) = u(x, kx − g(p)k) is a local diffeomorphism from (Rn , x0 )
into (S n , v0 ), where u(x0 ) = v0 and so that (v0 , p) ∈ Σm(k) (H).
Theorem 2.13. Let p ∈ M be an isolated non-flat rounding of g. Then indR (g, p) =
indF (ξ ◦ g, p), where the index of the flattening is considered with respect to the vector v0
and we choose the orientation in S n determined by u(x).
Proof. Note that
hu(x) (t) − hu(x) (p) = λ(x, t)(dx (t) − dx (p)),
for some function λ(x, t) > 0, where dx (t) is the distance-squared function of g and hv is
the height function of ξ ◦ g.
12
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
By taking local coordinates in M , we can assume that M = U , an open subset of Rm .
Then,
∂ |α| hu(x)
∂ |α| dx
(t)
=
λ(x,
t)
(t) +
∂tα
∂tα
X
λα,β (x, t)
1≤|β|<|α|
∂ |β| dx
∂ |α| λ
(t)
+
(x, t)(dx (t) − dx (p)),
∂tβ
∂tα
for some functions λα,β (x, t). Thus,
θF (ξ ◦ g, (u(x), t)) = Λ(x, t)θR (g, (x, t)) + A(x, t)(dx (t) − dx (p)),
where Λ(x, t) is a ν × ν matrix whose determinant is λ(x, t)ν > 0 and A(x, t) ∈ Rν . In
particular, this implies that the mapping degree of θF at (v0 , p) is equal to the mapping
degree of θR at (x0 , p).
¤
3. Curves in Rn
Let g : I → Rn be a smooth curve in Rn , where I ⊂ R is an interval. We have
that m = 1 and ν(m(k) ) = k, so that we can consider k-roundings with k = n + 1 and
k-flattenings with k = n. In fact, if g is given by g(t) = (g1 (t), . . . , gn (t)), then


g10 (t) g20 (t) · · · gn0 (t)
 00

 g1 (t) g200 (t) · · · gn00 (t) 
 000

000
000

F (g, t) = 
 g1 (t) g2 (t) · · · gn (t)  ,
..
.. 
 ..
...
.
. 
 .
(n)
(n)
(n)
gn (t)
g1 (t) g2 (t) · · ·
and t ∈ I is a flattening of g if and only if the determinant of the matrix F (g, t) is zero at
t. This definition agrees with the ordinary definition of flattening when g 0 (t), . . . , g (n−1) (t)
are linearly independent at any point t ∈ I, so that g has a well defined Frenet frame and
curvatures κ1 , . . . , κn−1 . In this case it is obvious that t ∈ I is a flattening if and only if
κn−1 (t) = 0.
We next set


g10 (t)
g20 (t)
···
gn0 (t)
0

 00
ϕ1 (t) 
···
gn00 (t)
g200 (t)
 g1 (t)

 000
000
000
,
(t)
ϕ
(t)
(t)
·
·
·
g
(t)
g
g
R(g, t) = 
2
n
2
1


..
..
..
.. 

..
.
.
.
.
. 

(n+1)
g1
(n+1)
(t) g2
(t) · · ·
(n+1)
gn
(t) ϕn (t)
where ϕk satisfy the following formula:
ϕ1 (t) = g10 (t)2 + · · · + gn0 (t)2 ,
ϕk (t) =
n
X
i=1
(k)
gi (t)gi0 (t) + ϕ0k−1 (t),
k = 2, . . . , n.
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
13
Then t ∈ I is a rounding of g if and only if the determinant of the matrix R(g, t) is zero
at t. Again, in the regular case, we can translate this condition in terms of the Frenet
frame e1 , . . . , en and the curvatures κ1 , . . . , κn−1 .
Definition 3.1. Let g : I → Rn be a smooth curve such that g 0 (t), . . . , g (n−1) (t) are
linearly independent at any point t ∈ I, and assume that it is parameterized by arc
length. At a point t ∈ I where κn−1 (t) 6= 0, the center of the osculating hypersphere is
the only point µ(t) ∈ Rn defined by the condition (µ(t), t) ∈ Σ1(n) (D).
The proof of the following result can be found in [28].
Proposition 3.2. Let g : I → Rn be as in Definition 3.1. Then,
µ = g + µ1 e1 + · · · + µn en ,
where µ1 = 0, µ2 = 1/κ1 , µ3 = µ02 /κ2 and
µk+1 = (µ0k + κk−1 µk−1 )/κk
for k = 3, . . . , n − 1.
Moreover, t is a rounding if and only if
µ0n (t) + κn−1 (t)µn−1 (t) = 0.
Example 3.3. If n = 2, then t ∈ I is a rounding if and only if µ02 (t) = 0, where µ2 = 1/κ.
Obviously, this is equivalent to κ0 (t) = 0, which is the ordinary definition of a vertex of a
regular plane curve. In fact, in this case it is possible to show that
det R(g, t) = −κ0 (t)kg 0 (t)k5 ,
which implies that t is a rounding if and only if it is a vertex, even when κ = 0.
The multiplicity of a flattening or a rounding is determined by the first non zero term in
the Taylor expansion of det F (g, t) or det R(g, t), respectively. For instance, if we suppose
that t0 = 0 is a flattening and det F (g, t) = ak tk + · · · , with ak 6= 0, then µF (g, 0) = k.
Example 3.4. Let g : R → R2 be the plane curve defined by t 7→ (t2 , t3 ). It is easy to see
that this curve has a flattening and a rounding at t = 0 with multiplicities µF (g, 0) = 2
and µR (g, 0) = 3.
We consider now the deformation gλ defined by gλ (t) = (t2 , (λ + t2 )t). The curvature
and its derivative are given by the following formulas:
κλ (t) =
2(3t2 − λ)
,
(4t2 + (λ + 3t2 )2 )3/2
and
κ0λ (t) =
24t(−9t4 + (3λ − 1)t2 + λ + 2λ2 )
.
(4t2 + (λ + 3t2 )2 )5/2
Thus gλ has two flattenings and three vertices near 0 if λ is a small positive number, and
gλ has no flattening points and one vertex near 0 if λ is a small negative number (see
Figure 1).
14
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
λ>0
λ=0
λ<0
Figure 1
Example 3.5. Let e0 , e1 , e2 be non-negative integers. We consider the plane curve g :
R → R2 defined by
g(t) = (te0 , te0 +e1 + ate0 +e1 +e2 ).
Then, by elementary computation, we have
det F (g, t) =e0 e1 (e0 + e1 )t2e0 +e1 −3 + · · · ,
det R(g, t) =e30 e1 (e0 + e1 )(e0 − e1 )t4e0 +e1 −6 + e0 e1 (e0 + e1 )3 (e0 + 2e1 )t4e0 +3e1 −6
+ ae30 (e0 − e1 − e2 )(e1 + e2 )(e0 + e1 + e2 )(e0 + 2e1 )t4e0 +e1 +e2 −6 + . . . .
In this case, the multiplicities are given by µF (g, 0) = 2e0 + e1 − 3, and


4e0 + e1 − 6 if e0 6= e1 ,




7e − 6
if e0 = e1 , 2e1 < e2 ,
0
µR (g, 0) =

7e0 − 6
if e0 = e1 , 2e1 = e2 , a 6= 1,




5e + e − 6 if e = e , 2e > e , a 6= 0.
0
2
0
1
1
2
The mapping degree of of det F (g, t) or det R(g, t) is equal, up to the sign, to the index
of an isolated flattening or rounding, respectively, whenever they are defined. However,
we have to be careful if we want to take into account the sign of the index. We will show
the following result just for roundings, where the sign has sense in any dimension.
Proposition 3.6. Let g : I → Rn be a smooth curve and let t0 ∈ I be an isolated non-flat
rounding. Then, the index indR (g, t0 ) is equal to the mapping degree of − det R(g, t) at
t0 .
Proof. The index indR (g, t0 ) is the mapping degree of θR Rn+1 , (x0 , t0 )) → (Rn+1 , 0) defined by
¢n+1
¡
θR (x, t) = hg (i) , x − gi − ϕi−1 i=1 ,
where x0 ∈ Rn is the only point such that θR (x0 , t0 ) = 0.
Since we suppose that t0 is a non-flat rounding, we have that g 0 , . . . , g (n+1) have rank
n at t0 . We will assume, for instance, that g 0 , . . . , g (n) are linearly independent (the other
cases should be similar). Then, we can take the following change of coordinates:
xi = hg (i) , x − gi − ϕi−1 ,
i = 1, . . . , n.
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
15
This coordinate change transforms θR into
θR (x, t) = (x, hg (n+1) , φ(x, t) − gi − ϕn ),
where φ(x, t) denotes the inverse of the coordinate change. In particular, this implies that
indR (g, t0 ) = sign(det(g 0 , . . . , g (n) )) deg(θR , (0, t0 )).
However we now have that the mapping degree of θR at (0, t0 ) is equal to the local degree
of hg (n+1) , φ(0, t) − gi − ϕn at t0 (since θR is an unfolding of this function). Thus, it only
remains to show that
det R(g, t) = − det(g 0 , . . . , g (n) )(hg (n+1) , φ(0, t) − gi − ϕn )).
Let us denote by Ri the (n + 1, i) minor
¯
¯ g0 · · ·
¯ 1
¯ .
Ri = ¯¯ ..
¯ (n)
¯g1
···
of R, that is,
gbi0
..
.
gn0
..
.
···
d
(n)
gi
···
(n)
gn
¯
¯
¯
¯
¯.
¯
¯
ϕn−1 ¯
0
..
.
It follows that
det R(g, t) = ϕn det(g 0 , . . . , g (n) ) +
n
X
(n+1)
(−1)n+i+1 gi
Ri .
i=1
On the other hand, φ(0, t) − g is determined by the linear system
hg (i) , φ(0, t) − gi = ϕi−1 ,
i = 1, . . . , n.
Thus, by the Cramer rule, we have that
φi (0, t) − gi = (−1)n+i
Ri
.
det(g 0 , . . . , g (n) )
In particular,
0
det R(g, t) = ϕn det(g , . . . , g
(n)
)−
n
X
(n+1)
gi
(φi (0, t) − gi ) det(g 0 , . . . , g (n) ),
i=1
which gives the desired result.
¤
4. Surfaces in R3
In this section, we suppose that g : U ⊂ R2 → R3 is a smooth map given by g(u, v) =
(g1 (u, v), g2 (u, v), g3 (u, v)) which defines a surface in R3 (possibly with singularities). Then
the first and the second fundamental forms for g are given by
I = Edu2 + 2F dudv + Gdv 2 ,
II = Ldu2 + 2M dudv + N dv 2
16
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
where
E = hgu , gu i,
F = hgu , gv i,
G = hgv , gv i,
det(gu , gv , guu )
det(gu , gv , guv )
det(gu , gv , gvv )
√
,
M= √
,
N= √
,
EG − F 2
EG − F 2
EG − F 2
and subscripts denote partial derivatives. Naturally, L, M, N are only defined if the
denominator does not vanish, that is, at the regular points of g. Umbilics are regular
points of the surface so that the second fundamental form is proportional to the first.
Thus we say that (u, v) is an umbilic of g if the rank of the matrix
Ã
!
E F G
L M N
L=
is less than 2. Moreover, it is well known that umbilic points of a regular surface correspond to points where the two principal curvatures (or eigenvalues of the second fundamental form) are equal.
In order to include the case where g may have singularities, we can define a new matrix
Ã
!
E F G
U (g, (u, v)) =
L0 M 0 N 0
where L0 = det(gu , gv , guu ), M 0 = det(gu , gv , guv ), N 0 = det(gu , gv , gvv ). This matrix
always makes sense. We observe that the rank of this matrix is not maximal either at an
umbilic or at a singular point of g.
4.1. 2-roundings of g. Now we look at 2-roundings (rounding, for short) of g. A rounding is a point where the rank of the matrix


g1 u g2 u g3 u 0
g

 1 v g2 v g3 v 0 


R(g, (u, v)) = g1 uu g2 uu g3 uu E 


g1 uv g2 uv g3 uv F 
g1 vv g2 vv g3 vv G
is less than four.
The following result is well-known (see Porteous [24]).
Proposition 4.1. A regular point of a smooth map g : U ⊂ R2 → R3 is a rounding if
and only if it is an umbilic. Moreover, it is non-flat if and only if the principal curvature
is not zero.
Now, we see that in the singular case, a rounding corresponds to a singular point which
is not of Whitney umbrella type. Remember that a singular point is a Whitney umbrella
if the map is generic in the Thom-Boardman sense (that is, its jet extension is transverse
to the Boardman strata).
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
17
Proposition 4.2. A singular point of a smooth map g : U ⊂ R2 → R3 is a rounding if
and only if it is not a Whitney umbrella.
Proof. If the map g has rank zero at (u0 , v0 ), it follows that it is always a rounding, but
it is not a Whitney umbrella (since the 1-jet extension of g is not transverse to Σ2 ).
Suppose now that g has rank one at (u0 , v0 ). Again we may assume that (u0 , v0 ) = (0, 0)
and g is given by
g(u, v) = (u, g2 (u, v), g3 (u, v)).
Then our matrix R(g, (u, v)) can be

1 g2 u g3 u
0 g
g3 v
2v


0 g2 uu g3 uu

0 g2 uv g3 uv
0 g2 vv g3 vv
written in the following form:

0

2
2


0 
E = 1 + g2 u + g3 u ,

E
F = g2 u g2 v + g3 u g3 v ,
 



F
G = g2 2v + g3 2v .
G
On the other hand, the map g defines a Whitney umbrella if and only if its 2-jet section
is transverse to the Σ1 strata. This holds if and only if
¯
¯
¯g
¯
¯ 2 uv g3 uv ¯
¯
¯ 6= 0 at 0.
¯g2 vv g3 vv ¯
This condition is equivalent to rank R(g, 0) = 4.
¤
In the last part of this section, we will study the index of a non-flat isolated rounding.
We have from Definition 2.10 that if p ∈ U is a non-flat isolated rounding of g : U ⊂
R2 → R3 , then there exists a unique x0 ∈ R3 such that (x0 , p) ∈ Σ2,2 (D). Then indR (g, p)
is equal to the mapping degree of the smooth map germ θR5 , (x0 , p)) → (R5 , 0) whose
components θi , i = 1, . . . , 5 are given by the following matrix equation:
 

  
0
θ1
g1 u g2 u g3 u  



θ   g
0
 2   1 v g2 v g3 v  x
   
  
θ3  = g1 uu g2 uu g3 uu  y  + E  .
 

  
F 
θ4  g1 uv g2 uv g3 uv  z
θ5
g1 vv g2 vv g3 vv
G
Let us denote by Pu , Pv , Puu , Puv , Pvv the five maximal minors of the matrix R(g, (u, v)).
That is,
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯ gu 0 ¯
¯ gu 0 ¯
¯ gu 0 ¯
¯ gu 0 ¯
¯ gv 0 ¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯g
¯
¯g
¯
¯g
¯
¯g
¯
¯g
0
0
0
E
E
¯
¯ v
¯
¯ v
¯
¯ v
¯
¯ uu
¯
¯ uu
,
P
=
,
P
=
,
P
=
,
P
=
Pu = ¯
¯.
¯
¯
¯
¯
¯
¯
¯
¯
vv
uv
uu
v
¯guu E ¯
¯guu E ¯
¯guv F ¯
¯guv F ¯
¯guv F ¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯guv F ¯
¯ gvv G¯
¯gvv G¯
¯ gvv G¯
¯ gvv G¯
18
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
We see that when the rounding is non-flat, then
C ∞ (R5 , (x0 , p))/hθ1 , . . . , θ5 i ∼
= C ∞ (R2 , p)/hPu , Pv , Puu , Puv , Pvv i.
This implies that we can choose a pair of elements P, Q ∈ {Pu , Pv , Puu , Puv , Pvv } which
generate the ideal R(g, p). Moreover, we have that the mapping degree of θ is equal, up
to sign, to the mappingl degree of (P, Q)R2 , p) → (R2 , 0).
The relations among Pu , Pv , Puu , Puv , Pvv are given in the following form:
(4.1)
(4.2)
gi u Pu − gi v Pv + gi uu Puu − gi uv Puv + gi vv Pvv = 0 (i = 1, 2, 3),
EPuu − F Puv + GPvv = 0.
4.2. Principal directions. We have another index at an isolated rounding, which is
associated with the principal directions of g. Suppose that (u, v) is a regular point which
is not umbilic. Then the principal directions of g at (u, v) are defined as the directions
determined by the eigenvectors of the second fundamental form at (u, v). Then we find
that the principal directions are given by the equation
¯
¯
¯ g1 u g2 u g3 u 0
¯
0
¯
¯
¯g
0 ¯¯
¯ 1 v g2 v g3 v 0
¯
¯
dv 2 ¯ = Puu dv 2 + Puv dudv + Pvv du2 = 0.
¯g1 uu g2 uu g3 uu E
¯
¯
¯g1 uv g2 uv g3 uv F −dudv ¯
¯
¯
¯g
2 ¯
g
g
G
du
1 vv
2 vv
3 vv
Thus, the principal directions define a pair of orthogonal line fields in the surface, which
are singular either at an umbilic or at a singular point of g. If p ∈ U is an isolated umbilic
or an isolated singular point of g, we denote by indP (g, p) the index at p associated with
any of these line fields. This is a half-integer and is equal to the number of turns of the
line field when we run through a small circle in U centered at p.
The classification of generic umbilics goes back to Darboux [5]. He found that there are
three types, namely, the lemon or D1 , the monstar or D2 and the star or D3 . Moreover,
he gave a description for the configuration of the curvature lines (integral curves of the
principal directions) in each case. In fact, it was shown by Gutierrez and Sotomayor
[10] (see also the work by Bruce and Fidal [3]) that g is principally structurally stable
at an umbilic if and only if it is one of the Darbouxian umbilics. By looking at these
configurations, it is easy to compute the index indP (g, p): D1 and D2 have index 1/2,
while D3 has −1/2.
The configuration of principal lines at a generic singular point can be found in [7]
(this is a corrected version of some erroneous results appearing in [11]). It follows that a
singular point of g is principally structurally stable if and only if it is a Whitney umbrella.
Moreover, from the description of principal lines at a Whitney umbrella we deduce that
the index indP (g, p) is 1/2.
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
19
In Figure 2, we present drawings for the configuration of principal lines at Darbouxian
umbilics D1 , D2 , D3 and at a Whitney umbrella W , obtained with the Superficies program
[21]. As a consequence, if the map g is not generic, we can take a generic deformation gλ
D1
D2
D3
W
Figure 2
and the index indP (g, p) is equal to (D1 + D2 − D3 + W )/2, where D1 , D2 , D3 , W denote
the number of umbilics of each type or Whitney umbrella that appear in gλ near p, for
λ 6= 0 small enough.
Moreover, the differential equation for principal lines can be seen as a particular case
of a positive quadratic differential (PQD) form in the sense of [9]. In local coordinates
(u, v), a PQD form is given by
ω = A(u, v)dv 2 + B(u, v)dudv + C(u, v)du2 ,
where A, B, C are smooth functions such that B 2 − 4AC ≥ 0 and B 2 − 4AC = 0 if and
only if A = B = C = 0. The points where A = B = C = 0 are called singular points of
ω. Associated with ω we have a pair of transversal line fields corresponding to the roots
of ω, which become singular precisely at the singular points of ω. Thus, if p is an isolated
singular point of ω, we can consider the index ind(ω, p) associated with any of these line
fields.
The following lemma, whose proof can be found in [13, Part 2, VIII, 2.3], shows that
the index of an isolated singular point of a PQD form is related to the mapping degree of
the coefficients.
Lemma 4.3. Let p be an isolated singular point of the PQD form ω = A(u, v)dv 2 +
B(u, v)dudv + C(u, v)du2 . Then,
1
1
ind(ω, p) = − deg((A, B), p) = − deg((B, C), p),
2
2
where deg((A, B), p) and deg((B, C), p) denote the mapping degrees of the maps (A, B)
and (B, C), respectively, at p.
As an immediate consequence, we get the following result, which allows us to compute
the index of the principal foliation in terms of the mapping degree of the coefficients
Puu , Puv , Pvv .
20
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
Corollary 4.4. Let g : U ⊂ R2 → R3 be a smooth map and let p ∈ U be either an isolated
umbilic or an isolated singular point of g. Then,
1
deg{(Puu , Puv )R2 , p) → (R2 , 0)}
2
1
= − deg{(Puv , Pvv )R2 , p) → (R2 , 0)}.
2
indP (g, p) = −
4.3. Indices of roundings. We will apply Corollary 4.4 to find relations between the
index defined by principal directions indP (g, p) and the index indR (g, p). However, we
have to distinguish cases depending on the rank of g at p.
4.3.1. Rank two case. We start by considering the case where p is a regular point.
Theorem 4.5. Let g : U ⊂ R2 → R3 be a smooth map and let p ∈ U be a non-flat isolated
umbilic of g with principal curvature k 6= 0. Then,
indP (g, p) =
1
sign(k) indR (g, p).
2
Proof. We will assume for simplicity that p = 0 is a non-flat umbilic and that g is given
by the Monge normal form g(u, v) = (u, v, w(u, v)). The matrix R(g, (u, v)) is equal to


1 0 wu
0
0 1 w
0 
v



2 ,
0 0 wuu 1 + wu 


0 0 wuv wu wv 
0 0 wvv 1 + wv2
and
Puv = wuu (1 + wv2 ) − wvv (1 + wu2 ),
Pvv = wuu wu wv − wuv (1 + wu2 ).
On the other hand, the index indR (g, p) is equal to the mapping degree of the map
germ θ given by



  
0
1 0 wu  
θ1
 0 
θ  0 1 w  x
v 


 2 

  
  
θ3  = 0 0 wuu  y  + 1 + wu2 



  
 wu wv 
θ4  0 0 wuv  z
1 + wv2
0 0 wvv
θ5
at the point P = (x0 , y0 , z0 , 0, 0) such that θ(x0 , y0 , z0 , 0, 0) = 0.
We can define new coordinates x̃, ỹ, z̃ by

  
  
0
x
1 0 wu
x̃

  
  
ỹ  = 0 1 wv  y  +  0  .
1 + wu2
z
0 0 wuu
z̃
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
21
Note that k 6= 0 implies wuu 6= 0. Moreover, sign(k) = sign(wuu ). Then, it follows that
deg θ = sign(k) deg θ̃, where θ̃ is the composite of θ with such coordinate change. A simple
computation gives that
µ
¶
wuv z̃ + Pvv wvv z̃ + Puv
θ̃(x̃, ỹ, z̃, u, v) = x̃, ỹ, z̃,
,
.
wuu
wuu
In particular, we get that
½µ
¶
¾
Pvv Puv
2
2
deg θ̃ = deg
,
R , 0) → (R , 0) = − deg{(Puv , Pvv )R2 , 0) → (R2 , 0)}.
wuu wuu
¤
Corollary 4.6. Let g : U ⊂ R2 → R3 be a smooth map and let p ∈ U be a non-flat isolated
(either regular or singular) rounding of g and let gλ : U → R3 be a generic deformation
of g, with λ ∈ (−², ²). Then,
indR (g, p) = D1+ − D1− + D2+ − D2− − D3+ + D3− ,
where Di+ , Di− denote the number of umbilics of type Di with positive or negative principal
curvature, respectively, that appear in gλ near p.
Remark 4.7. The classical Carathéodory conjecture states that every smooth convex
embedding of a 2-sphere in R3 must have at least two umbilics. This conjecture has
a stronger local version, known as the Loewner conjecture, which states that the index
indP (g, p) at any isolated umbilic g of a smooth regular surface g : U → R3 is always ≤ 1.
Since the sum of the indices of the umbilics of a compact immersed surface is equal to its
Euler-Poincaré characteristic (according to the Poincaré-Hopf formula) it follows that the
Loewner conjecture implies the Carathéodory conjecture, not only for a convex embedding
of a 2-sphere, but for any immersed compact surface M with χ(M ) 6= 0. The Loewner
conjecture is known to be true in the analytic case, although there is a big controversy
about the correct proof (see [31], [14]).
It would be interesting to see if indP (g, p) ≤ 1 is true or not in the rank one case
and provided that g is analytic. If this is true, we should obtain an extension of the
Carathéodory conjecture for surfaces with rank one singularities in R3 .
In the rank 0 case it is known that this conjecture is false. In fact, it is not difficult to
construct a surface with an isolated umbilic of index two. Here we present an example
which is inspired by the example given in [2]. Consider the map g : R2 → R3 given by
g(u, v) = (1/3)(−u3 − 3uv 2 , 3u2 v − v 3 , 3u4 − 3v 4 ). In Figure 3 we compute the principal
lines of this surface and show that it has an isolated umbilic of index two at the origin.
4.3.2. Rank one case. In the rank one case, the formulas of Corollary 4.4 are still true.
However, we have to be careful, because in this case, the index indP (g, p) not only counts
the number of umbilics that appear in a generic deformation, but this also includes the
number of Whitney umbrellas.
22
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
Figure 3
Example 4.8. Let us consider the smooth family of surfaces gλ : R2 → R3 defined by
gλ (u, v) = (u, v 2 , v(λ + v 2 − u2 )).
For λ = 0, g0 has an isolated singular point of rank one at 0. For λ > 0, gλ is generic
and has two Whitney umbrellas and two umbilics of type D3 near 0. For λ < 0, gλ is
regular and has no umbilics. Thus, it follows that the index indP (g, p) is equal to zero.
The configuration of the curvature lines near 0 is shown in Figure 4 in the three cases.
λ>0
λ=0
λ<0
Figure 4
In the rank one case, we can also obtain a formula to compute the index indR (g, p)
provided that the rounding is non-flat and isolated. However, it is not clear at all which
is the relation between this index and the index of the principal foliation indP (g, p).
Assume that p = 0 and that g is given in the form g(u, v) = (u, g2 (u, v), g3 (u, v)), where
g2 , g3 ∈ hu, vi2 . We have seen that g has a rounding at 0 if and only if it is not a Whitney
umbrella and it is non-flat if and only if the following matrix has rank two at 0:

g2 v g3 v

g
 2 uu g3 uu 
.

g2 uv g3 uv 
g2 vv g3 vv

ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
Let us denote the maximal minors of this matrix by
¯
¯
¯
¯
¯g
¯
¯g
¯
¯ 2 v g3 v ¯
¯ 2 v g3 v ¯
D12 = ¯
¯ , D13 = ¯
¯,
¯g2 uu g3 uu ¯
¯g2 uv g3 uv ¯
¯
¯
¯
¯
¯g
¯
¯g
¯
g
g
¯ 2
¯ 2 uu 3 uu ¯
3 uu ¯
D23 = ¯ uu
,
D
=
¯
¯
¯,
24
¯g2 uv g3 uv ¯
¯ g2 vv g3 vv ¯
¯
¯g
¯ 2
D14 = ¯ v
¯g2 vv
¯
¯g
¯ 2
D34 = ¯ uv
¯g2 vv
23
¯
g3 v ¯¯
¯,
g3 vv ¯
¯
g3 uv ¯¯
¯.
g3 vv ¯
It is obvious that D12 , D13 , D14 = 0 at 0. The fact that it is not a Whitney umbrella is
also equivalent to D34 = 0 at 0. Hence, it is non-flat if and only if either D23 6= 0 or
D24 6= 0.
Theorem 4.9. Let p = 0 ∈ U be an isolated non-flat rounding of rank one of the smooth
map g : U ⊂ R2 → R3 given by g(u, v) = (u, g2 (u, v), g3 (u, v)), with g2 , g3 ∈ hu, vi2 . Then,

− sign(D ) deg{(P , P )R2 , p) → (R2 , 0)}, if D 6= 0,
23
v
vv
23
indR (g, p) =
− sign(D24 ) deg{(Pv , Puv )R2 , p) → (R2 , 0)}, if D24 6= 0.
Proof. By definition, we have that indR (g, 0) = deg{θ : (R5 , 0) → (R5 , 0)}, where θ is
given by
  

 
θ1
1 g2 u g3 u  
0
θ  0 g



g3 v  x
2v
 2 
0
  
   
θ3  = 0 g2 uu g3 uu  y  + E  .
  

 
θ4  0 g2 uv g3 uv  z
F 
θ5
0 g2 vv g3 vv
G
Assume that D23 6= 0 at 0. We can define new coordinates x̃, ỹ, z̃ by
  
   
x̃
1 g2 u g3 u
x
0
  
   
ỹ  = 0 g2 uu g3 uu  y  + E  .
z̃
0 g2 uv g3 uv
z
F
Then, it follows that deg θ = sign(D23 ) deg θ̃, where θ̃ is the composite of θ with such a
coordinate change. A simple computation gives that
µ
¶
Pv
Pvv
+ h1 , ỹ, z̃,
+ h2 ,
θ̃(x̃, ỹ, z̃, u, v) = x̃,
D23
D23
where h1 , h2 ∈ hx̃, ỹ, z̃i. In particular, we get that
¶
¾
½µ
Pvv Pv
2
2
,
R , 0) → (R , 0) = − deg{(Pv , Pvv )R2 , 0) → (R2 , 0)}.
deg θ̃ = deg
D23 D23
The case D24 6= 0 is analogous.
¤
Remark 4.10. When D23 6= 0, we can use (4.1) and (4.2) in order to obtain Puv in terms
of Pv and Pvv . In fact, it follows that Puv = (D12 Pv + D24 Pvv )/D23 , which implies that
1
1
indP (g, p) = − deg{(Puv , Pvv )} = − sign(D23 ) deg{(D12 Pv , Pvv )}.
2
2
24
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
Analogously, if D24 6= 0, we get Puu = (D14 Pv + D34 Puv )/D24 , which gives
1
1
indP (g, p) = − deg{(Puu , Puv )} = − sign(D24 ) deg{(D14 Pv , Puv )}.
2
2
Note that D12 = D14 = 0 at 0, so that they have a relevant contribution to the mapping
degree of such maps.
4.3.3. Rank zero case. In the rank zero case, it is possible to obtain a similar result. Note
that a rank zero rounding of a map g : U ⊂ R2 → R3 is non-flat if and only if
¯
¯
¯
¯g
¯ 1 uu g1 uv g1 vv ¯
¯
¯
D = ¯g2 uu g2 uv g2 vv ¯
¯
¯
¯g3 uu g3 uv g3 uv ¯
is not zero at p. The proof of the following theorem is omitted, since it can be obtained
by using the same argument as in the proof of Theorem 4.9.
Theorem 4.11. Let g : U ⊂ R2 → R3 be a smooth map and let p ∈ U be an isolated
non-flat rounding of rank zero of g. Then
indR (g, p) = − sign(D) deg{(Pu , Pv )R2 , p) → (R2 , 0)}.
Example 4.12. Consider the map g : R2 → R3 defined by (u, v) 7→ (u2 , uv, v 2 ). This
map is generically two to one. If we take a deformation of this map, there is (at least)
one Whitney umbrella near 0. On the other hand, a simple computation shows that the
mapping degree of (Pu , Pv ) is −1 and hence indR (g, 0) = 1. We can conclude that there
exists at least one umbilic point near 0 in any deformation of g. On the other hand, it is
not difficult to see that the mapping degree of (Puv , Pvv ) is −2 so that indP (g, 0) = 1.
In fact, we can consider the generic deformation gλ (u, v) = (u2 + λv, uv, v 2 + λu). For
λ 6= 0 small enough, gλ has just one umbilic of type D1 with positive principal curvature
and one Whitney umbrella near the origin (see Figure 5.(a)). However, if we take a
different generic deformation gλ (u, v) = (u2 + λv, uv + λu − λv, v 2 − λu), this deformation
has one umbilic of type D3 with negative principal curvature and three Whitney umbrellas
(see Figure 5.(b)).
(a)
(b)
Figure 5
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
25
5. Complex coordinates
In this part, we discuss the possible values for the index of an isolated umbilic of a
regular surface in terms of the coefficients of the Monge normal form:
g = (g1 , g2 , g3 ) : R2 → R3 ,
(5.1)
(u, v) 7→ (u, v, w(u, v)).
To simplify notation, we identify R2 with C by introducing the complex coordinate z
defined by z = u + iv.
Theorem 5.1. We assume that 0 is an umbilic. Then we have
¾
½
³1
´
1
2
indP (g, 0) = − deg Φ : (C, 0) → (C, 0), z 7→ wzz
+ wz wz − wzz wz .
2
2
To prove this formula, we recall with some basic facts about complex coordinate. We
set z = u + iv, z = u − iv and
µ
¶
∂
∂
∂
∂
∂
∂
dz = du + idv, dz = du − idv,
=
+ ,
=i
−
.
∂u
∂z ∂z
∂v
∂z ∂z
Then, for any function f = f (u, v), we easily obtain the following relations:
 

 

fu
1 1
0 0 0
fz
f 
 i −i 0 0 0 
f 
 v

 z

 

 

1 2 1 .
fuu  = B fzz  , where B = 0 0
 

 

fuv 
0 0
fzz 
i 0 −i 
fzz
fvv
0
0
−1 2 −1
We can write down the first fundamental form in terms of z, and z in the following form:
I = (dg1 )2 + (dg2 )2 + (dg3 )2 = EC dz 2 + 2FC dzdz + GC dz 2 ,
where EC = g1 2z + g2 2z + g3 2z ,
consider the matrix

g1 z g2 z
g
 1 z g2 z

g1 zz g2 zz

g1 zz g2 zz
g1 zz g2 zz
and observe the
¯
¯ g1
¯ z
¯g
¯ 1
Pzz := ¯ z
¯g1 zz
¯
¯g1
zz
FC = g1 z g1 z + g2 z g2 z + g3 z g3 z , GC = g1 z2 + g2 2z + g3 z2 . Now we



g3 z 0
g1 u g2 u g3 u 0

g
g3 z 0 

 1 v g2 v g3 v 0 



g3 zz EC  = B −1 g1 uu g2 uu g3 uu E  ,



g1 uv g2 uv g3 uv F 
g3 zz FC 
g1 vv g2 vv g3 vv G
g3 zz GC
following relations:
¯
g2 z g3 z 0 ¯¯
F Puv − (E + G)Pvv + iEPuv
g2 z g3 z 0 ¯¯ Puu − Pvv − iPuv
=
.
¯=
16
16E
g2 zz g3 zz EC ¯
¯
g2
g3
FC ¯
zz
zz
The last equality we use (4.2).
26
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
Proof of Theorem 5.1. In the above formula, we have
Ã
!
1
F
E
Pzz =
(Puv , Pvv )
16E
−(E + G) 0
and we conclude that deg Pzz = deg(Puv , Pvv ), which allows us to compute the index of
an isolated umbilic by Theorem 4.5. We thus complete the proof, since
¯
¯
¯1/2 −i/2 wz
¯
0
¯
¯
¯1/2 i/2 w
¯
¶
µ
0
¯
¯
z
i
1
2
¯ =
wzz ( + wz wz ) − wzz wz .
Pzz = ¯¯ 0
¯
0
wzz
wz2
2
2
¯
¯
1
¯
¯
¯ 0
0
wzz
+ wz wz ¯
2
¤
Now we assume that
k
1
1
1
(5.2)
w = zz + C + Q +
U + higher order terms,
2
6
24
120
where k is the principal curvature at 0 and
C = αz 3 + 3βz 2 z + 3βzz 2 + α z 3 ,
Q = γz 4 + 4δz 3 z + 6cz 2 z 2 + 4δzz 3 + γ z 4 ,
and U is a homogeneous polynomial of degree five. Then we conclude that
1
1
Φ = Czz +
12
8
µ
1
Qzz − k 3 z 2
6
¶
1
+
4
µ
1
1
1
Uzz + k 2 zzCzz − k 2 zCz − k 2 z 2 Czz
30
6
3
¶
+ ···
and the Jacobian of Φ is equal to
1
|Φz |2 − |Φz |2 = (|α|2 − |β|2 ) + · · · .
4
Now we recover the following well-known fact by Theorem 5.1.
Proposition 5.2. If |α| > |β| (resp. |α| < |β|), then indP (g, 0) = −1/2 (resp. 1/2).
These cases correspond to the generic cases if
αz 3 + βz 2 z − βzz 2 − α z 3
has no multiple zeros (see [25, 12.4], [3, §2]).
Proposition 5.3. Consider the Monge normal form (5.1). We assume that
k
zz + H + higher order terms,
2
where H is a non-zero homogeneous polynomial of degree n with n ≥ 3. Then
w=
• If n ≥ 5, and k 6= 0, then indP (g, 0) = 1.
• If n ≥ 5, and if the map Hzz C, 0) → (C, 0) is finite, then indP (g, 0) ≤ 1.
• If n =
6 4, then | indP (g, 0)| ≤ (n − 2)/2.
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
27
Proof. We easily obtain that
1
1
Φ(z) = Hzz − k 3 z 2 + · · · ,
2
8
which shows the first assertion by Theorem 5.1. Since Hzz is not identically zero, we
may assume that the imaginary part of Hzz is not identically zero. We assume that Φ is
finite map. Then the image of oriented circle {z ∈ C : |z| = ε} cut the real axis at most
2(n − 2)-times, which implies | deg Φ| ≤ n − 2. This shows the last assertion.
We show the second assertion. We assume that Hzz C, 0) → (C, 0) is finite. Since
deg Hzz = deg Φ, it is enough to show that deg Hzz ≥ −2. We define a real-valued
function h(θ) by H(reiθ ) = rn h(θ). Let a+ (resp. a− ) denote the number of positive local
maximum (resp. minimum) of h. We easily see a+ ≥ a− . We define the map G by
G : (C, 0) −→ (C, 0),
reiθ 7−→ rn−2 ((n(n − 2)h(θ) − h00 (θ)) + i2(n − 1)h0 (θ)) .
Since Hzz = 14 e−2iθ G, we have deg Hzz = deg G−2 and it is enough to show that deg G ≥ 0.
Consider the set Z = S 1 ∩ G−1 {x ≥ 0, y = 0}. Each point of Z is a critical point of
h with n(n − 2)h − h00 > 0. Let b+ , b− denote the numbers of local maximums and
mimimums of h in the region n(n − 2)h − h00 > 0. Since a+ ≤ b+ , a− ≥ b− , we obtain
deg G = b+ − b− ≥ a+ − a− ≥ 0.
¤
Proposition 5.4. Consider the Monge normal form (5.1) with (5.2). Assume that C is
identically zero, and (γ, δ, c1 ) 6= (0, 0, 0) where c1 = c − k 3 /2. Set
¯
¯
¯ γ 2δ c1 0 ¯
¯
¯
¯ 0 γ 2δ c ¯
¯
1¯
R=¯
¯ = c1 4 − 2(|γ|2 + 2|δ|2 )c1 2 + 8<(γδ 2 )c1 + |γ|2 (|γ|2 − 4|δ|2 ).
¯c1 2δ γ 0 ¯
¯
¯
¯ 0 c1 2δ γ ¯
We choose θ1 so that γ1 = γe4θ1 i is a non-negative real number. Define real numbers p, q
by δ1 = δe−2θ1 i = p + qi. If 0 is an isolated umbilic, then we have the following:
(1)
(2)
(3)
(4)
(5)
| indP (g, 0)| ≤ 1.
If R > 0, |c1 | < γ1 (resp. |c1 | > γ1 ), then indP (g, 0) = −1 (resp. 1).
If R < 0, then indP (g, 0) = 0.
If R = 0, and c21 − γ12 6= 0 then | indP (g, 0)| ≤ 1/2.
If R = c21 − γ12 = 0, then indP (g, 0) = 0 except the following two cases:
(a) c1 − γ1 = 0, q = 0, |p| < |c1 |, (b) c1 + γ1 = 0, p = 0, |q| > |c1 |.
Proof. We easily obtain that
µ
¶
¢
1 1
1¡ 2
3 2
Qzz − k z + · · · =
γz + 2δzz + c1 z 2 + · · · ,
Φ(z) =
8 6
4
and Φ has rank zero at 0. If (γ, δ, c1 ) 6= (0, 0, 0), then the initial form is of degree 2, and
| deg Φ| ≤ 2. This implies 1.
28
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
Proof of 2, 3. Observe that R is the resultant of the initial form of Φ and Φ. If R 6= 0,
then the initial part is a homogeneous map of degree two with finite multiplicity. We
may assume that γ 6= 0. This implies that Φ is semi-quasi-homogeneous and the mapping
degree of Φ is equal to the mapping degree of its initial part (see [4]). Moreover, the initial
part has multiplicity 4 and according to [6], the mapping degree of Φ will be ±2 or 0.
Setting z = z1 eθ1 i , we have
Q =γ1 z14 + 4δ1 z13 z1 + 6cz12 z1 2 + 4δ1 z1 z1 3 + γ1 z1 4
1
R = (c21 − γ12 )2 − (c1 − γ1 )2 p2 − (c1 + γ1 )2 q 2 .
4
By looking at c1 , γ1 , p, q as coordinates in R4 , it is not difficult to see that the equation
R = 0 separates the region {γ1 > 0} of R4 in four connected components, corresponding
to the following regions: (1) R > 0, |c1 | < γ1 ; (2) R > 0, c1 > γ1 ; (3) R > 0, c1 < −γ1 and
(4) R < 0. Since the mapping degree will be constant on each one of this regions, it is
enough to compute the value of the mapping degree for a particular point (c1 , γ1 , p, q) in
each region. In Figure 6, we show the zero locus of R with γ1 = 1 in the (c1 , p, q)-space.
Figure 6
In the region (1), we take c1 = p = q = 0 and γ1 > 0. It follows that Φ = γ1 z 2 + · · · so
that the mapping degree of Φ is 2.
In the regions (2) and (3), we take γ1 = p = q = 0 and c1 6= 0, so that Φ = 14 c1 z 2 + · · ·
and the mapping degree is −2.
Finally, in the region R < 0, we take c1 = γ1 > 0, p = 0 and q 6= 0. Then Φ =
1
(γ (u2 − v 2 ), q(u2 + v 2 )) + · · · , which has mapping degree 0.
2 1
Proof of 4. We first assume that c21 − γ12 6= 0. Then the condition R = 0 is equivalent to
p2
q2
+
¡ c1 +γ1 ¢2 ¡ c1 −γ1 ¢2 = 1,
2
2
which defines a surface parameterized by
p=
c 1 + γ1
cos θ2 ,
2
q=
c1 − γ1
sin θ2 ,
2
0 ≤ θ2 ≤ 2π.
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
29
Then, setting z1 = u1 + v1 i, we obtain that
1
(c1 + γ1 )g1 − (c1 − γ1 )g2 i
Φ = (γ1 z12 + 2δ1 z1 z1 + c1 z1 2 ) + · · · =
g0 + · · · ,
4
2
where g0 = u1 cos θ22 + v1 sin θ22 , g1 = u1 cos θ22 − v1 sin θ22 , g2 = u1 sin θ22 + v1 cos θ22 . If we
know the image of the line {g0 = 0} by Φ, we can decide the mapping degree. Case by
case analysis shows that if Φ is finite, then | deg Φ| ≤ 1, and we complete the proof.
Proof of 5. Assume next that R = 0 and c21 − γ12 = 0. Then c1 − γ1 = 0 or c1 + γ1 = 0.
If c1 − γ1 = 0 (resp. c1 + γ1 = 0), then q = 0 (resp. p = 0) since R = 0. We then obtain
µ
µ
¶
¶
q 2
c1 + p 2 c1 − p 2
q 2
resp. Φ = − u1 + c1 u1 v1 + v1 i + · · · .
Φ=
u1 −
v1 + · · ·
2
2
2
2
The proof follows from the fact that the quadratic part of Φ is definite and, hence, the
mapping degree of Φ becomes zero.
¤
Remark 5.5. Assume that C is identically zero and γ = δ = c1 = 0. Then w is given by
w=
k3
k
zz − z 2 z 2 + H · · ·
2
8
where H is a non-zero homogeneous polynomial of degree n ≥ 5. If Φ = 21 Hzz + · · · is a
finite map, then 0 is an isolated umbilic, and (2 − n)/2 ≤ indP (g, 0) ≤ 1, by a discussion
similar to the proof of Proposition 5.3.
Example 5.6. We finish this section by showing that it is possible to construct surfaces
with umbilics of any index ≤ 1. Let n be an integer with n ≥ 2. Consider the surface
defined by w = (z n + z n )/2. Then we have
Φ(z) =
n(n − 1) n−2
z
+ ···
2
and indP (g, 0) = −(n − 2)/2.
Similarly we consider the surface defined by w = (zz)m (z n + z n )/2. Then
µ
¶
m(m − 1)
m−2 2
n
n
Φ(z) = (m + n)(m + n − 1)(zz)
z z +
z + ···
(m + n)(m + n − 1)
which is a map of mapping degree n − 2. Thus we get indP (g, 0) = −(n − 2)/2.
6. Surfaces in R4
In this last section, we will consider a smooth map g : U ⊂ R2 → R4 given by g(u, v) =
(g1 (u, v), g2 (u, v), g3 (u, v), g4 (u, v)), which defines a smooth surface in R4 (possibly with
singularities). We look at the 2-flattenings of g, which correspond to points where the
30
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
matrix


g1 u g2 u g3 u g4 u
g

 1 v g2 v g3 v g4 v 


F (g, t) = g1 uu g2 uu g3 uu g4 uu 


g1 uv g2 uv g3 uv g4 uv 
g1 vv g2 vv g3 vv g4 vv
has rank < 4.
On the other hand, the geometry of regular surfaces in R4 has been studied by several
authors. The local second-order invariants of these surfaces were described by Little [16]
in the following way: we have a first fundamental form
I = Edu2 + 2F dudv + Gdv 2 ,
E = hgu , gu i,
F = hgu , gv i,
G = hgv , gv i,
and two second fundamental forms
IIi = Li du2 + 2Mi dudv + Ni dv 2 ,
Li = hguu , ei i, Mi = hguv , ei i, Ni = hgvv , ei i, i = 1, 2,
being {e1 , e2 } an orthonormal basis of the normal plane to the surface at the point (this
makes sense only at the regular points of g). Associated with these quadratic forms we
have the following functions:
¯
¯
¯L1 2M1 N1
0 ¯¯
¯
1 ¯¯L2 2M2 N2
0 ¯¯
∆= ¯
K = L1 N1 − M12 + L2 N2 − M22 ,
¯,
¯
4¯0
L1 2M1 N1
¯
¯
¯0
L2 2M2 N2 ¯
and the matrix
Ã
!
L1 M1 N1
α=
.
L2 M2 N2
That is, ∆ is the resultant of the two second fundamental forms; K is called the Gaussian
curvature of g and it is shown in [16] that ∆, K and the rank of α are coordinate
independent. Then, we can classify the points of the surface in terms of these functions:
(1)
(2)
(3)
(4)
If ∆ < 0, the point is said to be hyperbolic.
If ∆ > 0, the point is said to be elliptic.
If ∆ = 0, the point is said to be parabolic.
If α has rank < 2, the point is said to be an inflection. Moreover, we have the
following types of inflection:
(a) if K > 0, it is an inflection of imaginary type;
(b) if K < 0, it is an inflection of real type;
(c) if K = 0, it is an inflection of flat type.
Proposition 6.1. A regular point of a smooth surface in R4 is a flattening if and only if
it is an inflection.
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
31
Proof. Since g is regular, we may assume that our point is the origin of R2 and that g is
given by g(u, v) = (u, v, g3 (u, v), g4 (u, v)), with
1
1
g4 = (a2 u2 + 2b2 uv + c2 v 2 ) + · · · .
g3 = (a1 u2 + 2b1 uv + c1 v 2 ) + · · · ,
2
2
Then, we have


1 0 0 0
0 1 0 0 
Ã
!


a1 b1 c1


.
F (g, 0) = 0 0 a1 a2  , and α(0) =


a2 b2 c2
0 0 b1 b2 
0 0 c1 c2
Here we take e1 = (0, 0, 1, 0) and e2 = (0, 0, 0, 1) as an orthonormal basis of the normal
plane to g at 0. The result is clear now.
¤
Remark 6.2. It follows from the above proof that if t is a regular point of g and the
matrix F (g, t) has rank < 3, then t is an inflection of flat type.
The multiplicity of a flattening µF (g, p) is related to the number of generic inflections
that appear in a generic deformation of g near p. Recall that it is defined by
µF (g, p) = dimR C ∞ (R2 , p)/F(g, p),
where F(g, p) is the ideal generated by Pu , Pv , Puu , Puv , Pvv , the maximal minors of F (g, t):
Pu = det(gv , guu , guv , gvv ),
Pv = det(gu , guu , guv , gvv ),
Puu = det(gu , gv , guv , gvv ),
Puv = det(gu , gv , guu , gvv ),
Pvv = det(gu , gv , guu , guv ).
Moreover, the relations between these five generators are given by:
gi u Pu − gi v Pv + gi uu Puu − gi uv Puv + gi vv Pvv = 0 (i = 1, 2, 3, 4).
When rank F (g, p) = 3, we can consider the index indF (g, p), which is defined as the
mapping degree of θ : (S 3 × R2 , (v0 , p)) → (R5 , 0) given by θ(v, t) = F (g, t)v and v0 ∈ S 3
is chosen so that (v0 , p) ∈ Σ2,2 (H). Since we have two possible choices for the vector,
namely ±v0 , the index is well defined up to sign. We have the following possibilities:
(1) Rank 2 case. Since g has rank 2, we see that gu , gv and one of {guu , guv , gvv }
are linearly independent at p. Suppose, for instance, that gu , gv , guu are linearly
independent. Then, Puv , Pvv generate the ideal F(g, p) and
| indF (g, p)| = | deg(Puv , Pvv )|.
(2) Rank 1 case. In this case one of {gu , gv } and two of {guu , guv , gvv } are linearly
independent at p. Suppose, for instance, that gu , guu , guv are linearly independent.
Then, Pv , Pvv generate the ideal F(g, p) and
| indF (g, p)| = | deg(Pv , Puv )|.
32
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
(3) Rank 0 case. Now the only possibility is that guu , guv , gvv are linearly independent
at p. Then, Pu , Pv generate the ideal F(g, p) and
| indF (g, p)| = | deg(Pu , Pv )|.
As in the case of a surface in R3 , the index indF (g, p) is related to an index defined by a
geometrical line field in the surface, namely the field of asymptotic directions. According
to [8], the asymptotic directions at a regular point are given by the quadratic differential
equation
¯
¯
¯ g1 u g2 u g3 u g4 u
0 ¯¯
¯
¯g
0 ¯¯
¯ 1 v g2 v g3 v g4 v
¯
¯
dv 2 ¯ = Puu dv 2 + Puv dudv + Pvv du2 = 0.
¯g1 uu g2 uu g3 uu g4 uu
¯
¯
¯g1 uv g2 uv g3 uv g4 uv −dudv ¯
¯
¯
¯g
g
g
g
du2 ¯
1 vv
2 vv
3 vv
4 vv
Moreover, the sign of the discriminant of this equation coincides with the sign of −∆.
Thus, at a hyperbolic point there exist exactly two asymptotic directions; at an elliptic
point there exist no asymptotic directions; at a parabolic point which is not an inflection,
there exists exactly one asymptotic direction and at an inflection, all the directions are
asymptotic.
Suppose now that p is an isolated inflection of imaginary type. This implies that ∆ = 0
and K > 0 at p and that ∆ < 0 for all t 6= p in a neighborhood of p. In particular,
we have a pair of asymptotic directions defined in such a neighborhood with an isolated
singularity at p. The index associated with these asymptotic directions will be denoted by
indA (g, p). The configuration of the asymptotic lines at a generic inflection of imaginary
type has been obtained recently in [8], where they show that the only possibilities are
again the Darbouxian configurations, with index ±1/2.
Note that if p is an inflection of imaginary type, then the condition K > 0 ensures that
both {gu , gv , guu } and {gu , gv , gvv } are linearly independent. Moreover, they determine
the same oriented 3-plane. That is, we have a canonical choice for the vector v0 ∈ S 3 such
that (v0 , p) ∈ Σ3,3 (H). This choice is given by
v0 =
gu ∧ gv ∧ guu
gu ∧ gv ∧ gvv
(p) =
(p).
kgu ∧ gv ∧ guu k
kgu ∧ gv ∧ gvv k
Thus, in the case of an inflection of imaginary type, we have a well defined index indF (g, p).
Theorem 6.3. Let g : U ⊂ R2 → R4 be a smooth map and let p ∈ U be an isolated
inflection of imaginary type of g. Then
1
1
1
indA (g, p) = − deg{(Puu , Puv )} = − deg{(Puv , Pvv )} = indF (g, p).
2
2
2
ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES
33
Proof. Since in this case both {gu , gv , guu } and {gu , gv , gvv } are linearly independent, both
{Puv , Pvv } and {Puu , Puv } generate the ideal F(g, p). Then, the two first equalities follow
from Lemma 4.3. To see the last equality, we will show that it is true for a generic
inflection. In the general case, we can show the result by taking a generic deformation of
g.
If p is a generic inflection, then the multiplicity µF (g, p) is one. Therefore, the map
germs (Puu , Puv )R2 , p) → (R2 , 0) and (Puv , Pvv )R2 , p) → (R2 , 0) are regular.
In general, if p is an inflection, we may assume that p = 0 and that g is given by the
Monge normal form g(u, v) = (u, v, g3 (u, v), g4 (u, v)), with
1
g3 (u, v) = (au2 + 2buv + cv 2 ) + · · · ,
2
1
g4 (u, v) = (pu3 + 3qu2 v + 3ruv 2 + sv 3 ) + · · · .
6
Then, Pu = Pv = 0 and
Puu = (br − cq)u + (bs − cr)v + · · · ,
Puv = (ar − cp)u + (as − cq)v + · · · ,
Pvv = (aq − bp)u + (ar − bq)v + · · ·
Note that if the inflection is of imaginary type, it follows that K = ac − b2 > 0. In
particular, ac > 0.
We compute now the Jacobian determinants of (Puu , Puv ) and (Puv , Pvv ) which give,
respectively, c∆ and a∆, with
∆ = a(r2 − qs) + b(ps − rq) + c(q 2 − rp).
Thus, the fact that the inflection is generic implies that ∆ 6= 0.
To see the value of indF (g, 0) we have to compute the Jacobian determinant of θ :
(S 3 × R2 , (v0 , 0)) → R5 given by θ(v, t) = F (g, t)v, where
v0 =
gu ∧ gv ∧ guu
(0).
kgu ∧ gv ∧ guu k
It is not difficult to see that if a > 0, then v0 = (0, 0, 0, 1) and the Jacobian determinant
of θ is −∆; if a < 0, then v0 = (0, 0, 0, −1) and the Jacobian determinant of θ is ∆. ¤
Remark 6.4. Note that an umbilic of a surface g in R3 is carried by the inverse of the
stereographic projection ξ : R3 → S 3 ,→ R4 into an inflection of the corresponding surface
ξ ◦ g in R4 (see the end of Section 2). Moreover, it follows from Theorem 3 of [18] that
the curvature lines of g are carried into the asymptotic lines of ξ ◦ g. Therefore, if p is an
isolated umbilic of g, then p is an inflection of imaginary type of ξ ◦ g and
indP (g, p) = indA (ξ ◦ g, p).
34
TOSHIZUMI FUKUI AND JUAN J. NUÑO-BALLESTEROS
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Department of Mathematics, Faculty of Science, Saitama University, 255 Shimo-Okubo,
Urawa 338-8570, Japan
E-mail address: [email protected]
Departament de Geometria i Topologia, Universitat de València, Campus de Burjassot,
46100 Burjassot SPAIN
E-mail address: [email protected]