Download 5. Surfaces with All Points Umbilic

Supplementary Notes for MM08 Geometry I
5. Surfaces with All Points Umbilic
Andrew Swann
P
roposition 6.5 in Pressley [1] describes those surfaces having the property that
every point is umbilic. The purpose of these notes is to prove this result with
more details regarding the global aspects. I will recall some of the pertinent definitions
before proving the main result.
uppose : U ! R3 is a patch for a regular surface
normal for this patch is
S
N = N
If
=
S R3.
The standard unit
u v
ku v k :
~ is a second patch then N~ = N on their common domain of definition.
Write
!
FI
E F
=
;
F G
FII
L M
=
;
M N
!
E = ku k2 ; F = hu ; v i ; G = kv k2 ;
L = huu ; Ni ; M = huv ; Ni ; N = hvv ; Ni
for the matrices of the first and second fundamental forms. The principal curvatures
are the roots 1 and 2 of the quadratic equation
det(FII
FI ) = 0 :
A point of S is an umbilic if 1 = 2 at that point. One can show that i are the
eigenvalues of the Weingarten matrix
W = FI 1FII :
Recall Proposition 6.4 shows that
Nu
Nv
!
=
WT
5.1
!
u
:
v
(5.1)
We know that f1 ; 2 g are the maximum and minimum normal curvatures at the
point. Choosing a patch with a different orientation changes the signs of the normal
curvatures and hence replaces f1 ; 2 g by f 1 ; 2 g. In particular, the Gaussian
curvature
K = 1 2
is independent of the choice of patch and so defines a smooth function
all of S .
K : S ! R on
e will need one topological lemma. A function f : X ! Y between topological
spaces is locally constant if for each x 2 X there is an open neighbourhood Ux
in X such that f is constant on Ux .
W
x
of
Lemma 5.1. Suppose
constant.
X is connected. If f : X
!Y
is locally constant, then
f is
S
Proof. First note that for any V Y , we have f 1 (V ) = x2f 1 (V ) Ux is a union of
open sets, so f 1 (V ) is open.
Fix x 2 X and let c = f (x). Consider the set A = f 1 (c) = f p 2 X : f (p) = c g
which is non-empty since it contains x. By our remark above, A is open. On the other
hand the complement X n A is f 1 (Y n fcg) so is also open. As X is connected it
follows that A = X and so f (p) = c for all p 2 X , i.e., f is constant.
T
he main theorem can now be stated and proved. Pressley’s proof concentrates on
the case of a single patch on S . In fact, his ideas go through directly if we know
the surface is oriented. Recall Proposition 4.5 says that S is oriented precisely when
there exists a globally defined unit normal field N. In any connected patch we have
either N = N or N = N .
Theorem 5.2. Suppose S is connected surface for which every point is umbilic.
Then either S is part of a sphere or S is part of a plane.
Proof. We start by proving that the Gaussian curvature K is constant.
For p 2 S , let (U ) be a connected patch around p. In (U ), the Weingarten matrix
is
!
W=
where
gives
0
;
0 = 1 = 2 is the common value of the principal curvatures. Equation (5.1)
Nu
= u ;
Nv
= v :
Considering the mixed second derivative of N we thus get
(u )v = (Nu )v = (Nv )u = (v )u :
5.2
(5.2)
Expanding this and cancelling the common term
uv , gives
v u = u v :
However the vectors u and v are linearly independent, so we must have
u = 0 = v
and is constant in (U ). This shows that K = 2 is locally constant and hence by
Lemma 5.1 K is constant.
There are now two cases: either K is identically zero or it is strictly positive. Let
us start with the case when K 0. In a given connected patch we have = 0
and so (5.2) implies that N is constant. Fix a vector v 2 R3 . Then the function
p 7! hN (p); vi2 is independent of the choice of and thus defines a global function
fv : S ! R. However, in the case K 0, the function fv is locally constant, so by
Lemma 5.1 fv is constant.
Choose v so that fv 6= 0. We may now define a global unit normal field N by
demanding that hN; vi > 0 at every point of S : if (U ) is a connected patch around q ,
we take N = sgn hN ; vi N . However, when K 0 we have seen that N and hence
N are locally constant. Therefore N is a constant vector, say w.
Using this global normal N = w, let (U ) be a patch on S . Then
h w ; i u = h w ; u i = h N; u i = 0 ;
h w ; i v = h w ; v i = h N; v i = 0
so the function p 7! hw; pi is locally constant on S . We thus have hw; pi = c with
c constant by Lemma 5.1, so each point of S lies in the plane f r : hw; ri = c g.
For the case K 6= 0, we may determine an orientation by choosing the charts that
p
have > 0. Let N be the resulting global unit normal field and note that = + K
is globally defined and constant. In a chart compatible with the orientation we have
(5.2) with constant, so the function p 7! N(p) + p is locally constant on S . By
Lemma 5.1 we have
N + p = a
for some constant vector a. As N(p) is a unit vector we have
p
2
1
1 2 1
1
= 2= ;
a = N(p)
K
for each point p of Sp
. We conclude that
and radius 1= = 1= K .
S is contained in the sphere with centre a=
5.3
References
[1] A. Pressley, Elementary differential geometry, Springer Undergraduate Mathematics Series, Springer-Verlag London Ltd., London, 2001.
Last revised: April 10, 2006.
5.4