A complete Riemannian metric on a manifold without boundary 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 each connected component (Burtscher 2025, pp. 76-77). 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 without boundary is complete.
