Smooth curves on a path connected smooth manifold

Question

Suppose that $M$ is a path connected smooth manifold, so any two points $p,q\in M$ can be joined with a continuous curve on $M$. Is it true that any two points can be joined with a smooth (I mean $C^{\infty}$) curve on $M$?

Answer 1

No need for a partition of unity. Let $\gamma$ be a path from $p$ to $q$, not necessarily smooth. Take finitely many local charts that cover the compact image of $\gamma([0,1])$. Call these local charts $U_1, \cdots, U_n$. Let us have ordered these charts so that the overlap between $U_{i}$ and $U_{i+1}$ is nonempty, but rather some point $r_i$. Pragmatically, you can do this by pulling back the $U_i$ to a cover of $[0,1]$ by the $\gamma^{-1}(U_i)$, and then ordering them by left-endpoints. Now, in each of these local charts the manifold looks like $\mathbb{R}^n$, so we can find a smooth path from $p$ to $r_1$ in $U_1$, then a smooth path from $r_1$ to $r_2$ in $U_2$, etc. Connecting all these smooth paths gives us a smooth path from $p$ to $q$.