# Metric on Riemannian manifolds

Why is it necessary to consider taking the infimum over the lengths of all piece-wise smooth curves while defining the distance function on a Riemannian Manifold instead of just taking the infimum over all smooth curves ? Are there examples where these two processes give different answers ?

• The distance function defined by infimum of paths lenght can be introduced for connected Riemannian manifolds.It is easy to see connectness is equivalent to arcwise connectness. Therefore we know that for any two points, say $p,q \in M$, there exists a CONTINUOUS path joining them . A priori this path is not smooth and not piecewise smooth. Using local coordinates it is always possible to construct a piecewise smooth path joining $p,q$ starting by the original continuous path. The question is: is it always possible to construct a smooth one? – user55449 Sep 21 '13 at 7:20