Hopf–Rinow theorem
From Infogalactic: the planetary knowledge core
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, who published it in 1931.[1]
Statement of the theorem
Let (M, g) be a connected Riemannian manifold. Then the following statements are equivalent:
- The closed and bounded subsets of M are compact;
- M is a complete metric space;
- M is geodesically complete; that is, for every p in M, the exponential map expp is defined on the entire tangent space TpM.
Furthermore, any one of the above implies that given any two points p and q in M, there exists a length minimizing geodesic connecting these two points (geodesics are in general critical points for the length functional, and may or may not be minima).
Variations and generalizations
- The Hopf–Rinow theorem is generalized to length-metric spaces the following way:
- If a length-metric space (M, d) is complete and locally compact then any two points in M can be connected by minimizing geodesic, and any bounded closed set in M is compact.
- The theorem does not hold in infinite dimensions: (Atkin 1975) showed that two points in an infinite dimensional complete Hilbert manifold need not be connected by a geodesic.[2]
- The theorem also does not generalize to Lorentzian manifolds: the Clifton–Pohl torus provides an example that is compact but not complete.[3]
Notes
References
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