In mathematics, a complete manifold (or geodesically complete manifold) is a (pseudo-) Riemannian manifold for which every maximal (inextendible) geodesic is defined on .
All compact manifolds and all homogeneous manifolds are geodesically complete.
Euclidean space , the spheres and the tori (with their usual Riemannian metrics) are all complete manifolds.
A simple example of a non-complete manifold is given by the punctured plane (with its usual metric). Geodesics going to the origin cannot be defined on the entire real line.
It can be shown that a finite dimensional path-connected Riemannian manifold is a complete metric space if and only if it is geodesically complete. This is the Hopf-Rinow theorem. This theorem does not hold for infinite dimensional manifolds. The example of a non-complete manifold (the punctured plane) given above fails to be geodesically complete because, although it is path-connected, it is not a complete metric space: any sequence in the plane converging to the origin is a non-converging Cauchy sequence in the punctured plane.