Geodesic manifold


In mathematics, a complete manifold is a Riemannian manifold for which, starting at any point, you can follow a "straight" line indefinitely along any direction. More formally, the exponential map at point, is defined on, the entire tangent space at.
Equivalently, consider a maximal geodesic. Here is an open interval of, and, because geodesics travel at fixed speed, it is uniquely defined up to translation. Because is maximal, maps the ends of to points of, and the length of measures the distance between those points. A manifold is geodesically complete if for any such geodesic, we have that.

Examples and non-examples

, the spheres, and the tori are all complete manifolds.
All compact Riemannian manifolds and all homogeneous manifolds are geodesically complete.
Every finite-dimensional path-connected Riemannian manifold which is also a complete metric space is geodesically complete. In fact, geodesic completeness and metric completeness are equivalent for these spaces. This is the content of the Hopf–Rinow theorem.

Non-examples

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 manifolds. It is the case for example of the Clifton–Pohl torus.
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.