A complete Riemannian metric is a Riemannian metric for which every geodesic extends to all real values of its affine parameter. This is equivalent to completeness of the induced distance on a manifold without boundary. A geodesic need not be an isometric copy of the real line; for example, great circle geodesics on the sphere are periodic. Euclidean space is complete, but the open unit disk is not complete since every nonconstant geodesic reaches the boundary at a finite distance. Whether or not a manifold is complete depends on the metric.
For instance, the punctured plane 
is not complete with the usual metric.
However, with the Riemannian metric 
, the punctured
plane is the infinite (flat) cylinder, which is
complete. The figure above illustrates a geodesic which
can only go a finite distance because it reaches a hole
in the punctured plane, exemplifying that the
punctured plane with the usual metric
is not complete. For the rescaled metric, the
path 
is a geodesic parametrized
by arc length.
Any Riemannian metric on a compact manifold without boundary is complete. Consequently, the pullback metric on the universal cover of a compact manifold is complete.
