Take Home Final Math 250A Fall 2010 1. Let M be a C 1 manifold and let X M . Let f 2 C 1 ( M ) be such that Xf = 0 and let t be the local ±ow generated by X . Then prove that for every p 2 M , f ( t ( p )) = f ( p ) : Use this observation to describe the leaves of the foliation of R 2 (0 ; 0) g corresponding to the subbundle of the tangent bundle given by p ! R X p with X = y 2 + x 2 : (Hint: Consider x 3 y 3 : ) 2. Let f : M ! R N be a proper embedding of a non-compact manifold M . Show that f ( M ) cannot be bounded. 3. Use the method of the proof we gave of the embedding theorem for compact manifolds to prove that a compact m-dimensional manifold with boundary can be embedded in R 2 m +1 . Use this result to prove that the interior of a compact m-dimensional manifold with boundary has a proper embedding into R 2 m +1 . 4. The purpose of this problem is to guide you through a proof of: If M is a connected C 1 manifold and p;q 2 M then there exists a di/eomorphism
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