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# LectureNotes10U - 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 10 2.3 Meaning of Gaussian Curvature In the previous lecture we gave a formal definition for Gaussian curvature K in terms of the differential of the gauss map, and also derived explicit formulas for K in local coordinates. In this lecture we explore the geometric meaning of K . 2.3.1 A measure for local convexity Let M R 3 be a regular embedded surface, p M , and H p be hyperplane passing through p which is parallel to T p M . We say that M is locally convex at p if there exists an open neighborhood V of p in M such that V lies on one side of H p . In this section we prove: Theorem 1. If K ( p ) > 0 then M is locally convex at p , and if k ( p ) < 0 then M is not locally convex at p . When K ( p ) = 0, we cannot in general draw a conclusion with regard to the local convexity of M at p as the following two exercises demonstrate: Exercise 2. Show that there exists a surface M and a point p M such that M is strictly locally convex at p ; however, K ( p ) = 0 ( Hint: Let M be the graph of the equation z = ( x 2 + y 2 ) 2 . Then may be covered by the Monge patch X ( u 1 , u 2 ) := ( u 1 , u 2 , (( u 1 ) 2 +( u 2 )) 2 ). Use the Monge Ampere equation derived in the previous lecture to compute the curvature at X (0 , 0).). Exercise 3. Let M be the Monkey saddle , i.e., the graph of the equation z = y 3 3 yx 2 , and p := (0 , 0 , 0). Show that K ( p ) = 0, but M is not locally convex at p . 1 Last revised: October 31, 2004 1

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After a rigid motion we may assume that p = (0 , 0 , 0) and T p M is the xy -plane. Then, using the inverse function theorem , it is easy to show that there exists a Monge Patch ( U, X ) centered at p , as the follwing exercise demonstrates: Exercise 4. Define π : M R 2 by π ( q ) := ( q 1 , q 2 , 0). Show that p is locally one-to-one. Then, by the inverse function theorem, it follows that π is a local diffeomorphism. So there exists a neighborhood U of (0 , 0) such that π 1 : U M is one-to-one and smooth. Let f ( u 1 , y 2 ) denote the z - coordinate of π 1 ( u 1 , u 2 ), and set X ( u 1 , u 2 ) := ( u 1 , u 2 , f ( u 1 , u 2 )). Show that ( U, X ) is a proper regular patch. The previous exercisle shows that local convexity of M at p depends on whether or not f changes sign in a neighborhood of the origin. To examine this we need to recall the Taylor’s formula for functions of two variables: f ( u 1 , u 2 ) = f (0 , 0) + 2 i =1 D i f (0 , 0) u i + 1 2 2 i,j =1 D ij ( ξ 1 , ξ 2 ) u i u j , where ( ξ 1 , ξ 2 ) is a point on the line connecting ( u 1 , u 2 ) to (0 , 0). Exercise 5. Prove the Taylor’s formula given above. ( Hints : First re- call Taylor’s formula for functions of one variable: g ( t ) = g (0) + g (0) t + (1 / 2) g ( s ) t 2 , where s [0 , t ]. Then define γ ( t ) := ( tu 1 , tu 2 ), set g ( t ) := f ( γ ( t )), and apply Taylor’s formula to g . Then chain rule will yield the desired result.) Next note that, by construction, f (0 , 0) = 0. Further D 1 f (0 , 0) = 0 = D 2 f (0 , 0) as well. Thus f ( u 1 , u 2 ) = 1 2 2 i,j =1 D ij ( ξ 1 , ξ 2 ) u i u j .
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LectureNotes10U - Math 497C Curves and Surfaces Fall 2004...

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