TITLE: Smooth curves on a path connected smooth manifold
QUESTION [16 upvotes]: 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 $M$?

REPLY [12 votes]: Since the accepted answer is rather incomplete, i'll post an answer here. As @Elchanan Solomon did, for any two points $p$ and $q$ on a connected manifold $M$, we can obtain a piecewise smooth curve $\gamma : [0,1] \to M$ such that $\gamma(0)=p$ and $\gamma(1)=q$. That is there are finite partition $0=t_0<t_1<\dots<t_{n-1}<t_n=1$ such that  $\gamma|_{[t_i,t_{i+1}]} : [t_i,t_{i+1}] \to M$ are smooth. To smoothing the curve $\gamma$ we can use smooth bump function (as the answer here) or we can use Whitney Approximation Theorem for Function.
For any point of corner $\gamma(t_i)$ where $\gamma$ not smooth, we choose a smooth chart $(U_i,\varphi)$ containing it. Now we pick a portion of $\gamma$ that still contain in $U_i$. That is, consider the restriction of $\gamma$ to a  closed interval $[a,b]$ where $a<t_i<b$ and $\gamma([a,b]) \subset U_i$. By composing with chart map $\varphi : U \to \mathbb{R}^n$, we obtain a piecewise smooth curve in $\mathbb{R}^n$,
$$
\alpha := \varphi \circ \gamma|_{[a,b]} : [a,b] \to \mathbb{R}^n.
$$
For a small number $\epsilon>0$, the subset $A=[a,t_i-\epsilon] \cup [t_i+\epsilon,b]$ is closed in $[a,b]$ and $\alpha$ is smooth on $A$. By Whitney Approximation theorem, there are smooth map $\tilde{\alpha} : [a,b] \to \mathbb{R}^n$ such that $\tilde{\alpha}$ is agree with $\alpha$ on $A$. Mapping back to $M$ and replacing the map $\varphi^{-1} \circ \tilde{\alpha} : [a,b] \to  M$ with $\gamma|_{[a,b]}$ before. Doing this for all corners, we obtain the desired smooth map joining $p$ and $q$.