This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 : ( , ) M with (0) = p and (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 2dimensional 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 suces 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 ) k D 1 X ( u 1 , u 2 ) D 2 X ( u 1 , u 2 ) k . 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 ) ....
View
Full
Document
This note was uploaded on 08/25/2011 for the course MATH 6456 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Staff
 Math, Geometry

Click to edit the document details