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 $\mathbb{R}$.

Examples

All compact Riemannian manifolds and all homogeneous manifolds are geodesically complete.

Euclidean space $\mathbb{R}^{n}$, the spheres $\mathbb{S}^{n}$ and the tori $\mathbb{T}^{n}$ (with their natural Riemannian metrics) are all complete manifolds.

A simple example of a non-complete manifold is given by the punctured plane $M := \mathbb{R}^{2} \setminus \{ 0 \}$ (with its induced metric). Geodesics going to the origin cannot be defined on the entire real line.

There exists non geodesically complete compact pseudo-Riemannian (but not Riemannian) manifolds. It is the case for example of the Clifton–Pohl torus.

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.