I Is There a Faster Way to Prove Smoothness on Manifolds?

[JonnyG]

So if $M, N$ are smooth manifolds then $F: M \rightarrow N$ is smooth if given $p \in M$, there is a smooth chart $(U, \phi)$ containing $p$ and a smooth chart $(V, \psi)$ containing $F(p)$ such that $\psi \circ F \circ \phi^{-1}: \phi(U \cap F^{-1}(V)) \rightarrow \mathbb{R}^n$ is smooth.

If I wanted to prove that a given function was smooth, are there any faster ways other than showing that its coordinate representation is smooth? For example, I just did a question where I had to show that $T(M \times N)$ is diffeomorphic to $T(M) \times T(N)$. I had to explicitly construct a bijection between the two manifolds then show that the coordinate representations of $F$ and $F^{-1}$ were smooth. This was a big pain. I wish there was a theorem I could have appealed to instead.

 

[micromass]

Inverse function theorem?
Sections of the tangent bundle?

It really depends on what you have seen already.

 

[JonnyG]

Thanks, micromass. I haven't learned the inverse function theorem on manifolds yet, but I suppose it's the usual inverse function theorem applied to the coordinate representation of the map I'm interested in. I am still early in my study of smooth manifolds - I'll be more patient.