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If $(M,g)$ is a Riemannian manifold, $g$ can be used to turn $M$ into a metric space.

For two points $p,q\in M$, $d(p,q)$ is usually defined to be the infimum of the lengths of all possible piecewise smooth (or $C^1$) curves connecting $p$ and $q$. So here's the question. Why is piecewise included in this definition? I suppose that one gets the same metric $d$ if one takes e.g. the infimum over all smooth curves.

I understand that it might me better to work over a larger class of curves for concrete computations but I wonder if there is another reason for this choice (maybe a deep reason I just don't see)? Also, in this case I think one might be able to further relax this regularity requirement?! Thanks for clarifications!

frog
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    We do not know that smooth curve attains a distance yet, but locally smooth curve attain a distance. – HK Lee Apr 06 '16 at 09:06

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One does get the same metric, since a piecewise smooth curve can be parametrized smoothly (using flat functions). But is seems more convenient to work with piecewise smooth curves instead. Just think about proving the triangle inequality ...

Andreas Cap
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  • So it's really just a technical thing you say? (Thanks actually for the comment about flat functions, I thought more about convolution but you're right of course)... – frog Apr 06 '16 at 09:34
  • I don't know about any deep reason. – Andreas Cap Apr 06 '16 at 09:52