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LectureNotes11U - Math 497C Curves and Surfaces Fall 2004,...

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Math 497C Mar 3, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 11 2.4 Intrinsic Metric and Isometries of Surfaces Let M R 3 be a regular embedded surface and p , q M , then we deFne dist M ( p, q ) := inf { Length[ γ ] | γ :[0 , 1] M, γ (0) = p, γ (1) = q } . Exercise 1. Show that ( M, dist M ) is a metric space. Lemma 2. Show that if M is a C 1 surface, and X M is compact, then for every ²> 0 , there exists δ> 0 such that ¯ ¯ dist M ( p, q ) −k p q k ¯ ¯ ² k p q k for all p , q X , with dist M ( p, q ) δ . Proof. DeFne F : M × M R by F ( p, q ) := dist M ( p, q ) / k p q k ,i f p 6 = q and F ( p, q ) := 1 otherwise. We claim that F is continuous. To see this let p i be a sequnce of points of M which converge to a point p M. We may assume that p i are contained in a Monge patch of M centered at p given by X ( u 1 ,u 2 )=( u 1 2 ,h ( u 1 2 )) . Let x i and y i be the x and y coorindates of p i .I f p i is sufficiently close to p =(0 , 0), then, since h (0 , 0) = 0, we can make sure that k∇ h ( tx i ,ty i ) k 2 ², for all t [0 , 1] and 0. Let γ , 1] R 3 be the curve given by γ ( t tx i i ( tx i i )) . 1 Last revised: November 8, 2004 1
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Then, since γ is a curve on M , dist M ( p, p i ) Length[ γ ] = Z 1 0 q x 2 i + y 2 i + h∇ h ( tx i ,ty i ) , ( x i ,y i ) i 2 dt Z 1 0 q x 2 i + y 2 i + ² ( x 2 i + y 2 i ) 2 dt 1+ ² q x 2 i + y 2 i (1 + ² ) k p p i k So, for any ²> 0wehave 1 dist M ( p, p i ) k p p i k ² provided that p i is sufficiently close to p . We conclude then that F is con- tinuous. So U := F 1 ([1 , ² ]) is an open subset of M × M which contains the diagonal ∆ M := { ( p, p ) | p M } . Since ∆ X M is compact, we may then choose δ so small that V δ (∆ X ) U , where V δ (∆ X ) denotes the open neighborhood of ∆ X in M × M which consists of all pairs of points ( p, q ) with dist M ( p, q ) δ .
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This note was uploaded on 08/25/2011 for the course MATH 6456 taught by Professor Staff during the Spring '08 term at Georgia Tech.

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LectureNotes11U - Math 497C Curves and Surfaces Fall 2004,...

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