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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.

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

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I don't have enough reputation to comment, so I just put it as an answer:

Your question is answered e.g. here: Link

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  • $\begingroup$ Yeah but my question is about finding a smooth function with this property, the answer you showed me just find a continuous, even piece-wise smooth but not smooth that is what I want. Thanks anyway. $\endgroup$ – kvicente Jul 7 '17 at 19:26
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    $\begingroup$ @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 $\mathbb{R}^n$.. $\endgroup$ – Bettybel Jul 7 '17 at 19:28
  • $\begingroup$ How do I precisely smooth out, that´s the main issue, is there any procedure to do this? $\endgroup$ – kvicente Jul 7 '17 at 19:29
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    $\begingroup$ @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. $\endgroup$ – Bettybel Jul 7 '17 at 19:32
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    $\begingroup$ I see, thanks for pointing that out. I still think that one should be able to find the answer to this on the web with not much effort. For example, take a look at the accepted answer here: Link $\endgroup$ – user12390 Jul 7 '17 at 19:33

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