Hopf–Rinow theorem
From Wikipedia, the free encyclopedia
(Redirected from Hopf-Rinow theorem)
In 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.
[edit] Statement of the theorem
Let (M, g) be a 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 extrema, and may or may not be minima). This does not hold in infinite dimensions: (Atkin 1975) showed that two points in an infinite dimensional complete Riemannian manifold need not be connected by any geodesic.
[edit] Generalization
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.
[edit] References
- Atkin, C. J. (1975), "The Hopf-Rinow theorem is false in infinite dimensions", The Bulletin of the London Mathematical Society 7 (3): 261–266, doi:, MR0400283, http://blms.oxfordjournals.org/cgi/reprint/7/3/261.pdf
- Hopf, H., Rinow, W., Über den Begriff der vollständigen differentialgeometrischen Fläche, Comment. Math. Helv. 3 (1931), 209-225.
- Jürgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin. ISBN 3-540-42627-2See section 1.4.
- Voitsekhovskii, M.I. (2001), "Hopf–Rinow theorem", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
