Hopf–Rinow theorem

The set of Hopf Rinow is a central message of the Riemannian geometry. He states that coincide on Riemannian manifolds, the terms of the geodesic completeness and the completeness in the sense of metric spaces. Named the sentence after the mathematicians Heinz Hopf and his student Willi Rinow.

Geodetic full diversity

A connected Riemannian manifold is geodetically complete if the exponential is defined for all for all. That is, for each point and each tangent vector is the geodesic with and defined on all of.

Set of Hopf and Rinow

Let be a finite, connected Riemannian manifold. Then the following properties are equivalent:

From these four equivalent statements can be another reason.

• For all, there exists a geodesic connecting the points on the shortest path. The distance function is defined as the minimum over all piecewise differentiable curves with and, which means it applies For this function, the defining properties of a metric apply.

Examples

• It follows from the theorem of Hopf Rinow that all compact, connected Riemannian manifolds ( geodesically ) complete. For example, the sphere completely.
• The Euclidean space and hyperbolic space are also complete.
• The metric space with the metric induced by the Euclidean standard scalar is not complete. If one chooses namely a point, so there's no point to the shortest link in.