TITLE: Metric on Riemannian manifolds
QUESTION [7 upvotes]: 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 ?

REPLY [3 votes]: The two infima are the same. Just for some proofs it is convenient to consider 
piecewise-smooth paths.