Assignment 1

# Assignment 1 - PMATH 465/665 Riemannian Geometry Assignment...

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PMATH 465/665: Riemannian Geometry Assignment 1; due Tuesday, 24 September 2013 Note: All manifolds are always smooth in this course. Unless noted otherwise, all tensors (functions, vector fields, forms, etc.) are also always smooth , as are all maps between manifolds. [1] Show that every vector field X with compact support is complete. That is, show that the integral curve of X through any p M is defined for all t R . [2] Let Θ be a flow on an oriented manifold. Show that for each t R , the diffeomorphism Θ t is orientation-preserving wherever it is defined. [3] Let M be a connected manifold. Show that the group of diffeomorphisms of M acts transitively on M . More precisely, for any two points p, q M , show that there is a diffeomorphism F : M M such that F ( p ) = q . [ Hint: First prove the following lemma: If p, q lie in the open unit ball B of R n , there is a compactly supported vector field on B whose flow Θ satisfies Θ 1 ( p ) = q . ] [4] For each k -tuple of vector fields on R 3 shown below, either find local coordinates ( u 1
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