Given $f : P\rightarrow N$ $C^\infty$ and $\pi : M\rightarrow N$ local diffeomorphism show that $\tilde f$ s.t.$f= \tilde f \circ \pi$ is $C^\infty$

Question

Let be $M$, $N$ and $P$ three differentiable manifolds.

I consider $\pi: M \rightarrow N$ a local diffeomorphism and $f:P \rightarrow N$ differentiable.

I have to prove that the application $$ \tilde f :P\rightarrow M$$ such that $$f= \tilde f \circ \pi$$ is differentiable.

Since $\pi$ is a local diffeorphism I know that its differential is iniective in every point of $M$, but $M$ is not a submanifold since $\pi$ is not necessarily iniective. I tried to work locally with neighbourhoods and restrictions, since the property of being $C^\infty$ is a local property but I don't know how to proceede with the details and I'm stuck. Should I use something like the implicite function theorem?

Any help would be greatly appreciated. Thanks!

Answer 1

Locally $\pi$ is a diffeomorphism, so restricted to sufficiently small neighborhoods we have $\tilde f = f \circ \pi^{-1}$. Then $\tilde f$ is differentiable with derivative given by the chain rule if such a $\tilde f$ exists. In general such a $\tilde f$ may not exist, i.e. if $\pi$ is the map $\Bbb R \rightarrow S^1$ given by $\pi(t)=e^{it}$ and $f$ is the identity on $S^1$.