When working on Riemannian manifolds, it is commonly assumed that the manifold is complete and compact. (The case of non-compactness is also of interest too.) In this entry, let us review the role of completeness and compactness in this setting.

**How important the completeness is?** Let us recall that for given a Riemannian manifold , what we have is a nice structure as well as an appropriate analysis on any tangent space . For a -curve on , the length of is

where a tangent vector. (Note that by using curves, the tangent vector is being understood as follows

for any differentiable function at .) Length of piecewise curves can be defined as the sum of the lengths of its pieces. From this a distance on whose topology coincides with the old one on is given as follows

where the infimum is taken on all over the set of all piecewise curves connecting and .