e1_m7350_s2011_v1_0 - Exam 1: Math 7350 Spring 2011 Problem...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Exam 1: Math 7350 Spring 2011 Problem 1. (15) (a) (10) Let M be a smooth manifold and let S M be an embedded submanifold. Prove that S is a closed subset of M if and only if the inclusion map ι : S , M is a proper map. (Recall that if X and Y are topological spaces and F : X Y is a map, then F is proper if F - 1 ( C ) is compact for every compact set C Y .) (b) (5) Give an example of a smooth manifold M and an embedded submanifold S M such that S is not closed in M . Problem 2. (10) Let M and N be smooth manifolds. If F : M N is a smooth map and c N is such that F - 1 ( c ) is an embedded submanifold of M with codim( F - 1 ( c )) = dim( N ), must c be a regular value of F ? Either prove this or give a counterexample. Problem 3. (10) (a) (5) Let M be a smooth manifold and let S M be an embedded submanifold. Suppose γ : J M is a smooth curve such that γ ( J ) S . Prove that for every t J , γ 0 ( t ) is in the subspace T γ ( t ) S of T γ ( t ) M .
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/03/2011 for the course MATH 3364 taught by Professor Staff during the Spring '08 term at University of Houston.

Ask a homework question - tutors are online