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Suppose $\mathcal{M}$ is a smooth, connected manifold, and let $p$ and $q$ be distinct points in $\mathcal{M}$. Is it true that there always exists a smooth injective regular map $\gamma: [0,1] \rightarrow \mathcal{M}$ such that $\gamma(0)=p$ and $\gamma(1)=q$?

Rahul Sarkar
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  • Do you mean path connected instead of connected? – Jean-Armand Moroni Oct 04 '22 at 07:03
  • What have you tried? Wouldn't it be enough to solve the problem for the case when $p$ and $q$ lie in a coordinate neighborhood? – Laz Oct 04 '22 at 07:04
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    @Jean-ArmandMoroni, path connectedness and connectedness are equivalent on manifolds. – Laz Oct 04 '22 at 07:07
  • @Laz I tried the following: if I can have a finite number of coordinate neighborhoods $a_1,a_2,...,a_k$ such that only successive ones intersect (so they form like a "chain"), and $p \in a_1$, $q \in a_k$, then I can do it by using piecewise straight lines in each chart. The total curve is then injective. Then all kinks can be smoothed out at the end. But I dont know how to form this chain of coordinate charts. – Rahul Sarkar Oct 04 '22 at 19:24
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    I would do something similar. By reducing to the case $\mathbb{R}^n$, the smoothening trick should be a lot easier. To get the finite chain of coordinate neighborhoods "connecting" the points, take any continuous curve $\gamma: I\rightarrow M$ joining $p$ and $q$, and notice that $\gamma(I)$ is compact. – Laz Oct 04 '22 at 20:16
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    oh wow, thanks! that was easy. – Rahul Sarkar Oct 05 '22 at 20:34

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