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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