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Vector fields

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Advanced Geometry 2017–2018 ENS de Lyon

Vector fields

Exercise 1(Flow box). LetM be a compact manifold and let X be a vector field onM. Let p ∈ M be such that X(p) 6= 0, prove that there exist local coordinates around p such that X= ∂x∂

1.

Are the orbits ofX submanifolds of M?

Exercise 2 (Transitivity of the group of diffeomorphisms). 1. Letaandbin the open ball B = {x ∈Rn | kxk< 1}. Prove that there exists a diffeomorphism f :Rn → Rn such that f(a) =band f = idon Rn\B.

2. LetM be a connected manifold, prove that it is path connected.

3. If moreoverdim(M)>2, prove that the natural action of the group of diffeomorphisms of M isk-transitive for any k∈N∗.

4. What happens ifdim(M) = 1?

1

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