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
.