Given a smooth compact $n$-dimensional manifold $M^{n}$, let $\operatorname{Diff}(M)$ denote the group of smooth diffeomorphisms $M \rightarrow M$ equipped with the Whitney $C^{\infty}$-topology. Let $h_{\phi} \colon [0, 1] \times M \rightarrow M$ denote the homotopy associated to a path $\phi \colon [0, 1] \rightarrow \operatorname{Diff}(M)$. My question is: Given such $\phi$, how can one prove that there exists a path $\psi \colon [0, 1] \rightarrow \operatorname{Diff}(M)$ with the same endpoints as $\phi$ and such that $h_{\psi}$ is smooth?

So far, I have neither succeeded in finding a formal proof nor a detailed reference in the literature. I am particularly interested in the case where $M$ has non-empty boundary. I am looking forward to helpful comments!

  • $\begingroup$ You would want some sort of Whitney approximation theorem that improves smoothness in the $t$-direction while remaining at least $C^1$-close in the $M$-direction. I believe such a thing, though I don't know if it's been written down anywher. $\endgroup$ – user98602 Mar 3 '16 at 15:49

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