Geodesic manifold
In mathematics, a complete manifold (or geodesically complete manifold) is a (pseudo-) Riemannian manifold for which every maximal (inextendible) geodesic is defined on .
Examples
All compact manifolds and all homogeneous manifolds are geodesically complete.
Euclidean space , the spheres
and the tori
(with their natural Riemannian metrics) are all complete manifolds.
A simple example of a non-complete manifold is given by the punctured plane (with its induced metric). Geodesics going to the origin cannot be defined on the entire real line.
Path-connectedness, completeness and geodesic completeness
It can be shown that a finite-dimensional path-connected Riemannian manifold is a complete metric space (with respect to the Riemannian distance) 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.
References
- O'Neill, Barrett (1983), Semi-Riemannian Geometry, Academic Press, ISBN 0-12-526740-1. See chapter 3, pp. 68.