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LectureNotes9U - Math 497C Curves and Surfaces Fall 2004...

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Math 497C Oct 7, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 9 2.2 Definition of Gaussian Curvature Let M R 3 be a regular embedded surface, as we defined in the previous lecture, and let p M . By the tangent space of M at p , denoted by T p M , we mean the set of all vectors v in R 3 such that for each vector v there exists a smooth curve γ : ( , ) M with γ (0) = p and γ (0) = v . Exercise 1. Let H R 3 be a plane. Show that, for all p H , T p H is the plane parallel to H which passes through the origin. Exercise 2. Prove that, for all p M , T p M is a 2-dimensional vector sub- space of R 3 ( Hint: Let ( U, X ) be a proper regular patch centered at p , i.e., X (0 , 0) = p . Recall that dX (0 , 0) is a linear map and has rank 2. Thus it suﬃces to show that T p M = dX (0 , 0) ( R 2 )). Exercise 3. Prove that D 1 X (0 , 0) and D 2 X (0 , 0) form a basis for T p M ( Hint: Show that D 1 X (0 , 0) = dX (0 , 0) (1 , 0) and D 2 X (0 , 0) = dX (0 , 0) (0 , 1)). By a local gauss map of M centered at p we mean a pair ( V, n ) where V is an open neighborhood of p in M and n : V S 2 is a continuous mapping such that n ( p ) is orthogonal to T p M for all p M . For a more explicit formulation, let ( U, X ) be a proper regular patch centered at p , and define N : U S 2 by N ( u 1 , u 2 ) := D 1 X ( u 1 , u 2 ) × D 2 X ( u 1 , u 2 ) D 1 X ( u 1 , u 2 ) × D 2 X ( u 1 , u 2 ) . Set V := X ( U ), and recall that, since ( U, X ) is proper, V is open in M . Now define n : V S 2 by n ( p ) := N X 1 ( p ) . 1 Last revised: October 8, 2004 1

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Exercise 4. Check that ( V, n ) is indeed a local gauss map.
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LectureNotes9U - Math 497C Curves and Surfaces Fall 2004...

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