1

Let $M$ be a connected smooth manifold of dimension $m\geq 2$, and let $\{p_1,\cdots,p_n\}$ and $\{q_1,\cdots,q_n\}$ be two set of points in $M$.

(a) Prove: there exists a diffeomorphism $\Phi$ such that $\Phi(p_i)=q_i$ holds for all $i$.

My attempts

I can reduce the question to finding a $\Phi$ such that $\Phi(p)=q$. To give that, I have proved the following statement (in one word, vector fields can be extended)

Let $M$ be a connected smooth manifold and $A\subset M$ is a closed subset. Let $X$ be a smooth vector field on $A$, then for any open subset $U$ with $A\subset U$, there exists a (global) smooth vector field $\widetilde{X}$ such that $\widetilde{X}|_{A}=X$ and $\text{supp}(\widetilde{X})\subset U$.

Finally, if I can find a smooth vector field such that some curve connecting $p$ and $q$ is an integral curve, then a diffeomorphism may be constructed and can be extended to $M$.

Isomorphism
  • 437
  • 2
  • 10
  • 2
    What tools do you know for constructing diffeomorphisms of manifolds? What did you try to solve these problems? The way you wrote these questions, they appear to be homework problems, so why would anybody solve them for you? – Moishe Kohan Dec 01 '21 at 14:26
  • Partitions of unit may be of help. – DavideL Dec 01 '21 at 14:27
  • Partition of unity is used in the extension of the vector field. @DavideL – Isomorphism Dec 01 '21 at 14:57
  • I have done some preparations and reduced it to a single point case. The next step in my opinion is to solve is for some nice $M$, like $\mathbb{B}^{m}$. But my mind got black thinking about the construction and what should be done after that.@MoisheKohan – Isomorphism Dec 01 '21 at 15:04

0 Answers0