SCIENCE CHINA
Mathematics
. ARTICLES .
January 2010 Vol. 53 No. 1: 251–256
doi: 10.1007/s11425-010-0004-z
On surfaces immersed in Euclidean space R4
Dedicated to Professor Yang Lo on the Occasion of his 70th Birthday
PENG ChiaKuei1 & TANG ZiZhou2,∗
1School
of Mathematical Sciences, Graduate University, Chinese Academy of Sciences, Beijing 100049, China;
2School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems,
Beijing Normal University, Beijing 100875, China
Email: [email protected], [email protected]
Received December 4, 2008; accepted February 2, 2009
Abstract Let M be a closed oriented surface immersed in R4 . Associated it one has the generalized Gauss map
from M into the Grassmann manifold G4,2 . This note will be concerned with the geometry of the generalized
Gauss map by using the moving frame theory and the quaternion interpretation of Plücker coordinates. As one
of consequences, we get the celebrated theorem of Chern and Spanier, Hoffman and Osserman, who proved it
by quite different methods. At last, we give an explicit construction of a series of immersions of S 2 in R4 with
any given normal Euler number.
Keywords
surface, normal euler number, quaternion multiplication
MSC(2000):
53A05, 53C42
Citation: Peng C K, Tang Z Z. On surfaces immersed in Euclidean space R4 . Sci China Math, 2010, 53(1): 251–256,
doi: 10.1007/s11425-010-0004-z
1
Plücker coordinates and quaternion multiplication
This section derives an interpretation of Plücker coordinates in view of quaternion multiplication, which
will bring a big advantage for the theory of submanifold geometry.
Let G4,2 denote the Grassmann manifold of oriented planes through the origin of the Euclidean space
4
R of four dimension. The Plücker coordinates will give rise to an explicit homeomorphism from G4,2 to
the topological product S 2 × S 2 of two standard 2-spheres in the Euclidean space R3 . Thus composing
two projections to factors respectively, one has two maps:
G, G : G4,2 −→ S 2
(1.1)
where we write G(P ) = (x1 , x2 , x3 ) ∈ S 2 and G (P ) = (y1 , y2 , y3 ) ∈ S 2 for P ∈ G4,2 .
The explicit constructions of the maps G and G are given as follows. Following Chern and Spanier [2],
let ε1 , ε2 , ε3 , ε4 be an (fixed) oriented orthonormal basis of R4 such that ε1 ∧ ε2 ∧ ε3 ∧ ε4 coincides with
the standard orientation of R4 . For any given P in G4,2 , let e1 , e2 be an orthonormal basis of P such
that e1 ∧ e2 agrees with the orientation of P . Then
e1 ∧ e2 = a12 ε1 ∧ ε2 + a23 ε2 ∧ ε3 + a31 ε3 ∧ ε1 + a34 ε3 ∧ ε4 + a14 ε1 ∧ ε4 + a24 ε2 ∧ ε4 .
∗ Corresponding
(1.2)
author
c Science China Press and Springer-Verlag Berlin Heidelberg 2010
math.scichina.com
www.springerlink.com
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Clearly these Plücker coordinates aij of P are independent of the choice of e1 , e2 and satisfy the
following two equalities:
a12 a34 + a23 a14 + a31 a24 = 0,
(1.3)
a212 + a234 + a223 + a214 + a231 + a224 = 1.
Conversely, any set of six real numbers satisfying (1.3) must be the Plücker coordinates for some oriented
plane in R4 . Furthermore we introduce a linear change of coordinates by
x1 = a12 + a34 , x2 = a23 + a14 , x3 = a31 + a24 ,
y1 = a12 − a34 , y2 = a23 − a14 , y3 = a31 − a24 .
(1.4)
Therefore G4,2 is homeomorphic to the subset S 2 × S 2 of R6 consisting of (x1 , x2 , x3 , y1 , y2 , y3 ) such that
2 2
xi = yi = 1.
Fortunately we can identify R4 with the algebra of quaternions, and identify S 2 with the unit quaternions with zero real part. In this way we define two maps Φ, Φ : G4,2 −→ S 2 by
Φ(P ) = e2 e1 ,
Φ (P ) = e2 e1 ,
(1.5)
where e1 , e2 are positively oriented orthonormal basis of P , and e1 denotes the conjugation of the quaternion e1 . Since e1 , e2 = 0, it follows that e1 e1 , e2 e1 = 0, which ensures e2 e1 ∈ S 2 . Similarly
e2 e1 ∈ S 2 . Moreover it is clear to see that e2 e1 , e2 e1 ∈ S 2 are independent of the choice of e1 , e2 .
Therefore the maps Φ and Φ are well-defined. The relation of G (G ) and Φ (Φ ) is described as the
following
Lemma 1.1. Φ = F ◦ G, Φ = F ◦ G , where F and F are two diffeomorphisms of S 2 preserving the
orientation defined by F (x, y, z) = (x, −z, y), F (x, y, z) = (−x, z, y) for (x, y, z) ∈ S 2 .
Proof.
Let P = span(e1 , e2 ) with e1 = a1 + a2 i + a3 j + a4 k, e2 = b1 + b2 i + b3 j + b4 k. Then
Φ(P ) = (b1 + b2 i + b3 j + b4 k)(a1 − a2 i − a3 j − a4 k) = (a12 + a34 )i + (a13 + a42 )j + (a23 + a14 )k
= x1 i − x3 j + x2 k
and thus as claimed Φ = F ◦ G. The proof of Φ = F ◦ G is similar.
The proof is now complete.
We conclude this section with some interesting and elementary formulas of quaternions which will be
useful in next section.
Lemma 1.2.
Let e1 , e2 , e3 , e4 be a positively oriented orthonormal basis of R4 . Then the following
equalities hold:
(1) e1 e2 e3 e4 = 1, e1 e2 e3 e4 = −1;
(2) e3 e1 = e2 e4 , e4 e1 = −e2 e3 , e4 e1 = e2 e3 , e3 e1 = −e2 e4 .
Proof. (1) We will only prove e1 e2 e3 e4 = 1, and leave the proof of the formula e1 e2 e3 e4 = −1
to the reader. First consider the special case that e1 = 1. Let e2 = a1 i + a2 j + a3 k, e3 = b1 i + b2 j + b3 k,
e4 = c1 i + c2 j + c3 k with
⎞
⎛
a1 a2 a3
⎟
⎜
⎜ b1 b2 b3 ⎟ ∈ SO(3).
(1.6)
⎠
⎝
c1 c2 c3
Then
e1 e2 e3 e4 = {c1 (a2 b3 − a3 b2 ) + c2 (a3 b1 − a1 b3 ) + c3 (a1 b2 − a2 b1 )}
+ {c1 (a1 b1 + a2 b2 + a3 b3 ) + c2 (a1 b2 − a2 b1 ) + c3 (a1 b3 − a3 b1 )}i
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+ {c1 (a2 b1 − a1 b2 ) + c2 (a1 b1 + a2 b2 + a3 b3 ) + c3 (a2 b3 − a3 b2 )}j
+ {c1 (a3 b1 − a1 b3 ) + c2 (a3 b2 − a2 b3 ) + c3 (a1 b1 + a2 b2 + a3 b3 )}k
= 1,
where the last equality follows from (1.6).
Next consider a general positively oriented orthonormal basis e1 , e2 , e3 , e4 . Since 1 = e1 e1 , e1 e2 ,
e1 e3 , e1 e4 is still a positively oriented orthonormal basis of R4 , we get
1 = (e1 e2 ) (e1 e3 ) (e1 e4 ) = e1 e2 e3 e4 .
(2)
e3 e1 = −e1 e3 = −e1 (e1 e2 e3 e4 ) e3 = −e2 e3 (e4 e3 ) = e2 e3 e3 e4 = e2 e4 .
The proofs of the left 3 equalities are similar and will be omitted.
2
Oriented surfaces immersed in R4
Throughout this section, let M be a closed oriented surface smoothly immersed in R4 . By mapping each
point p of M to the oriented plane through the origin parallel to the tangent plane to M at p, we obtain
a map T : M −→ G4,2 , the generalized Gauss map of M . Composing the projections G and G in (1.1)
respectively, we have two maps
g, g : M −→ S 2 .
(2.1)
ϕ = Φ ◦ T, ϕ = Φ ◦ T : M −→ S 2 .
(2.2)
Using (1.5) to define
It is easy to see that ϕ = F ◦ g, ϕ = F ◦ g . The main purpose of this note is to study the differentials
dϕ and dϕ in terms of the Gaussian curvature and normal curvature of the immersion.
Given any point p in a Riemannian manifold N n , around p we choose a local field of orthonormal
frames e1 , . . . , en . Set
ωA = eA , dp,
ωAB = deA , eB , 1 A, B n.
These will give the structure equations ensued
dp = ωA eA ,
(2.3)
deA = ωAB eB , ωAB + ωBA = 0;
dωA = ωAB ∧ ωB ,
dωAB = ωAC ∧ ωCB + ΩAB , ΩAB + ΩBA = 0,
(2.4)
where ΩAB (eB , eA ) = Sec(eA ∧ eB ) is the sectional curvature of the plane spanned by eA and eB .
Now let M 2 −→ R4 be an immersion of a closed oriented surface M . To each point p ∈ M , we choose
a local field of positively oriented orthonormal frames e1 , e2 , e3 , e4 such that e1 and e2 are tangent to M
at p. Restricted to M , one has the following Gauss equation and Ricci equation according to (2.4):
⎧
⎨dω12 = ω13 ∧ ω32 + ω14 ∧ ω42 ,
⎩
dω34 = ω31 ∧ ω14 + ω32 ∧ ω24 .
(2.5)
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PENG ChiaKuei et al.
Theorem 2.1.
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At any point p ∈ M , we have
det(dϕ)p = K(p) + KN (p),
det(dϕ )p = K(p) − KN (p),
where K(p) = −dω12 (e1 , e2 )p , KN (p) = −dω34 (e1 , e2 )p are the Gaussian curvature and normal curvature
respectively.
Proof.
By (2.2), (1.5) and (2.3) we find
dϕ = d(e2 e1 ) = (de2 )e1 + e2 (de1 ) = ω23 e3 e1 + ω24 e4 e1 + ω13 e2 e3 + ω14 e2 e4 .
Observing that by lemma1.2, we can set E1 = e3 e1 = e2 e4 , E2 = e4 e1 = −e2 e3 . Thus the above
arguments yield
(2.6)
dϕ = (ω14 + ω23 ) E1 + (ω24 − ω13 ) E2 .
Furthermore, one can verify that
Ei , Ej = δij , Ei , ϕ = 0.
(2.7)
Hence E1 and E2 is a local field of orthonormal frames of S 2 , and (ω14 + ω23 ) ∧ (ω24 − ω13 ) is the pullback
of the volume element of S 2 . Consequently we arrive at
det(dϕ)p = (ω14 + ω23 ) ∧ (ω24 − ω13 )(e1 , e2 )p = −(dω12 + dω34 )(e1 , e2 )p
by using (2.5). The proof of det(dϕ )p = K(p) − KN (p) is similar and is omitted.
It completes the proof of Theorem 2.1.
Remark.
As the referee pointed out that, the results of Theorem 2.1 were first derived by Hoffman
and Osserman [4]. To prove it, they identified the Grassmannian of oriented two-planes in R4 with the
quadric in CP 3 , and then used the local conformal parameter and complex function theory.
A direct consequence of Theorem 2.1 is the celebrated theorem of Chern and Spanier [2]. They proved
it by methods of algebraic topology.
Corollary 2.2.
Let M be a closed oriented surface immersed in R4 . Then the mapping degree of g
and g are given by
deg g =
1
(χ(M ) + χ(ν)),
2
deg g =
1
(χ(M ) − χ(ν)),
2
where χ(ν) denotes the Euler number of the normal bundle ν.
Proof.
By the definition (2.2), integrating over M we deduce immediately
1
1
1
deg g =deg ϕ =
det(dϕ) =
(K(p) + KN (p)) = (χ(M ) + χ(ν)).
4π M
4π M
2
The proof of deg g = 12 (χ(M ) − χ(ν)) is similar.
Remark.
When M is an embedding oriented surface, χ(ν) vanishes. Thus in this case we establish
deg g = deg g = 12 χ(M ) as Chern and Spanier stated.
Corollary 2.3.
Let M be an immersed surface in R4 with constant normal curvature. Then there
exists at least a point p0 with non-negative Gaussian curvature.
Proof. Suppose this is not true, the Gaussian curvature K is negative everywhere. Then the GaussBonnet formula implies that χ(M ) is negative. On the other hand, by Theorem 2.1 we find that, the
tangent map of one of ϕ and ϕ has negative determinant everywhere. Hence either ϕ : M −→ S 2 or
ϕ : M −→ S 2 is a submersion. Since M is closed, a submersion must be a covering. Furthermore since
S 2 is simply connected, it follows that either ϕ or ϕ is a diffeomorphism. As a result, χ(M ) = 2, a
contradiction. The proof is now complete.
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Remark.
The assumption that normal curvature is constant is equivalent to that the area of the
ellipse of curvature is constant (see [3, 5]).
Corollary 2.4.
Let M be an immersed surface in S 3 (1). Then there exists at least a point p0 with
non-negative Gaussian curvature.
Corollary 2.5. Let M be an immersed surface in R4 with positive Gaussian curvature. Then deg g =
degg = 1, where g, g : M ∼
= S 2 −→ S 2 .
Proof. The assumption that the Gaussian curvature is always positive guarantees that the mean curvature normal vector field H never vanishes (use the Gauss equation [7]). Thus the normal bundle of the
immersion admits a cross section without zeros. Therefore χ(ν) = 0. The conclusion we expected will be
followed from Corollary 2.2 and the Gauss-Bonnet formula. It completes the proof.
3
Immersed spheres in R4
A famous theorem of Smale [6] states that, two immersions from S 2k into R4k are regularly homotopic if
and only if their normal Euler numbers are equal. Furthermore, any even number can be the normal Euler
number of some immersion S 2k −→ R4k . The main aim of this section is to give an explicit construction
of a series of immersions from S 2 into R4 with all possible normal Euler numbers (even numbers). The
computation of the normal Euler number is heavily dependent on our theory developed in last section.
First we need to construct a “standard” continuous map S 2 −→ S 2 which has degree n for n 1. Let
ψn : S 2 −→ S 2 be given by
ψn (cos θ cos ϕ, cos θ sin ϕ, sin θ) = (cos θ cos nϕ, cos θ sin nϕ, sin θ),
(3.1)
where a point (different from the north and south poles) of S 2 is represented by the coordinates (cos θ cos ϕ,
cos θ sin ϕ, sin θ) for − π2 < θ < π2 , 0 < ϕ 2π. Clearly one can extend the definition of ψn at the north
and south poles so that ψn is continuous. We have as expected
Lemma 3.1.
Proof.
n.
deg ψn = n.
ψn is just the suspension of the map S 1 −→ S 1 sending eiϕ to einϕ , which has of course degree
Theorem 3.2.
Let fn : S 2 −→ R4 = C 2 (n 1) be given by
1
z
n
fn (x, y, z) =
(x + iy) ,
(x + iy),
1 + z2
n
for (x, y, z) ∈ S 2 ⊂ R3 . Then fn is an immersion with normal Euler number 2n.
Proof.
The proof is divided into three steps.
Step 1. The verification that fn is an immersion. First observe that fn is immersive around both the
north and south poles. Next for the points where z = ±1, we use the coordinates
x = cos θ cos ϕ,
y = cos θ sin ϕ,
z = sin θ.
In this way, fn is rewritten as
fn = (α cos ϕ, α sin ϕ, β cos nϕ, β sin nϕ),
where
α=
Since
cos θ
,
1 + sin2 θ
β=
cosn θ sin θ
.
n(1 + sin2 θ)
⎧
∂fn
⎪
⎨
= (−α sin ϕ, α cos ϕ, −nβ sin nϕ, nβ cos nϕ),
∂ϕ
⎪
⎩ ∂fn = (α̇ cos ϕ, α̇ sin ϕ, β̇ cos nϕ, β̇ sin nϕ),
∂θ
(3.2)
(3.3)
(3.4)
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PENG ChiaKuei et al.
where
α̇ =
It follows that
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cosn−1 θ(1 − (n + 2) sin2 θ − (n − 1) sin4 θ)
− sin θ(3 − sin2 θ)
,
β̇
=
.
2
(1 + sin θ)2
n(1 + sin2 θ)2
∂fn ∂fn
,
∂ϕ ∂θ
= 0,
∂fn ∂ϕ > 0 and
(3.5)
∂fn ∂θ > 0 whenever |θ| < π/2.
Step 2. We choose two orthonormal tangent vectors at the points where z = ±1:
⎧
∂fn
⎪
1
∂ϕ
⎪
⎪
e
= (1 + cos2n−2 θ sin2 θ)− 2 (− sin ϕ, cos ϕ, − cosn−1 θ sin θ sin nϕ, cosn−1 θ sin θ cos nϕ),
=
⎪ 1
⎨
n
| ∂f
|
∂ϕ
∂fn
⎪
⎪
⎪
2
2 −1
∂θ
⎪
⎩e2 = ∂fn = (α̇ + β̇ ) 2 (α̇ cos ϕ, α̇ sin ϕ, β̇ cos nϕ, β̇ sin nϕ).
| ∂θ |
It gives
e2 ē1 =h(θ)(0, −(α̇ + β̇ cosn−1 θ sin θ), (α̇ cosn−1 θ sin θ − β̇) sin(n + 1)ϕ,
− (α̇ cosn−1 θ sin θ − β̇) cos(n + 1)ϕ).
1
where h(θ) = ((α̇2 + β̇ 2 )(1 + cos2n−2 θ) sin2 θ)− 2 .
Thus we have a well-defined (even at the north and south poles) map ϕ = e2 ē1 : S 2 −→ S 2 , which is
clearly homotopic to a map ψ : S 2 −→ S 2 that is defined by
ψ(cos θ cos ϕ, cos θ sin ϕ, sin θ)
= h(θ)((β̇ − α̇ cosn−1 θ sin θ) cos(n + 1)ϕ, (β̇ − α̇ cosn−1 θ sin θ) sin(n + 1)ϕ, −(α̇ + β̇ cosn−1 θ sin θ)).
Step 3. The computation of deg ψ. A simple observation is that ψ + ψn+1 = 0, where ψn+1 is given
in (3.1). Hence ψ is homotopic to ψn+1 , which has degree n + 1 by Lemma 3.1. Consequently
deg ϕ = deg ψ = deg ψn+1 = n + 1.
On the other hand, by Corollary 2.2, we have deg ϕ =
as hoped χ(ν) = 2n.
The proof is now complete.
Remark 1.
1
2
(χ(S 2 ) + χ(ν)) = 1 + 12 χ(ν). Therefore we get
For n = 1, f1 : S −→ R in Theorem 3.2 is just the well-known Whitney sphere.
2
4
Remark 2.
Composing a diffeomorphism R4 −→ R4 which reverses the orientation with fn (n 1),
we get an immersion S 2 −→ R4 with normal Euler number −2n.
Acknowledgements The project was partially supported by National Natural Science Foundation of China
(Grant Nos. 10531090 and 10229101) and the Chang Jiang Scholars Program.
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