Complete manifold

In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, there are straight paths extending infinitely in all directions.

[1] A geodesic is maximal if its domain cannot be extended.

is (geodesically) complete if for all points

[1] The Hopf–Rinow theorem gives alternative characterizations of completeness.

is (geodesically) complete if and only if it satisfies one of the following equivalent conditions:[2] Euclidean space

(with their natural Riemannian metrics) are all complete manifolds.

All symmetric spaces are geodesically complete.

A simple example of a non-complete manifold is given by the punctured plane

Geodesics going to the origin cannot be defined on the entire real line.

By the Hopf–Rinow theorem, we can alternatively observe that 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.

There exist non-geodesically complete compact pseudo-Riemannian (but not Riemannian) manifolds.

In the theory of general relativity, which describes gravity in terms of a pseudo-Riemannian geometry, many important examples of geodesically incomplete spaces arise, e.g. non-rotating uncharged black-holes or cosmologies with a Big Bang.

The fact that such incompleteness is fairly generic in general relativity is shown in the Penrose–Hawking singularity theorems.

is geodesically complete, then it is not isometric to an open proper submanifold of any other Riemannian manifold.