I Simpler way to prove smoothness

1. Jan 31, 2017

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.

2. Feb 1, 2017

micromass

Staff Emeritus
Inverse function theorem?
Sections of the tangent bundle?

It really depends on what you have seen already.

3. Feb 2, 2017

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.