Geodesic completeness

A Riemannian manifold is geodesically complete when every maximal geodesic is defined for every real affine parameter, equivalently when every exponential map is defined on its entire tangent space.

For a connected Riemannian manifold, metric completeness, geodesic completeness, compactness of every closed bounded set, and the existence of a minimizing geodesic between every pair of points are equivalent.