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Hopf-rinow TheoremIn mathematics, the Hopf-Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow. The theorem is stated as follows: Let M be a Riemann manifold. Then the following statements are equivalent: - The closed and bounded subsets of M are compact.
- M is a complete metric space
- M is a complete topological space
- M is geodesically complete; that is, for every p in M, the exponential map is defined on the entire tangent space .
Furthermore, any one of the above implies that given any two points p and q in M, there exists a geodesic connecting these two points, and that furthermore, this geodesic is of minimal length (geodesics are in general extrema, and may or may not be minima). References - Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin. ISBN 3-540-4267-2 See section 1.4.
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