# Construction of a Smooth Curve on a Connected Manifold [closed]

Does anyone know how to prove the existence of a smooth curve $c$ from the unit interval of the real numbers to a connected smooth manifold $M$ such that $c(0)=p$ and $c(1)=q$ with $p$ and $q$ points on $M$ ? This is a question from Spivak´s Differential Geometry Vol 1. Thanks for the help.

## closed as off-topic by José Carlos Santos, A. Goodier, Lazy Lee, Dap, agtortorellaMar 29 '18 at 21:07

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• @kvicente Once you have the continuous path, cover it with finitely many charts (since it is compact) and smooth out on each chart since each chart is just an $\mathbb{R}^n$.. – Bettybel Jul 7 '17 at 19:28
• @kvicente I can assume the charts are just balls from $\mathbb{R}^n$. Deep inside the balls (like radius minus epsilon) you can replace the path with straight lines if you like by joining the first entry point to the last entry point. In the intersection of charts use flat functions (the $e^{-1/x^2}$ business) to join together the line segments from each chart. – Bettybel Jul 7 '17 at 19:32