Hopf-Rinow Theorem
Get Hopf%E2%80%93Rinow Theorem essential facts below. View Videos or join the Hopf%E2%80%93Rinow Theorem discussion. Add Hopf%E2%80%93Rinow Theorem to your PopFlock.com topic list for future reference or share this resource on social media.
Hopf%E2%80%93Rinow Theorem

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 (Mg) be a connected Riemannian manifold. Then the following statements are equivalent:

  1. The closed and bounded subsets of M are compact;
  2. M is a complete metric space;
  3. 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

Notes

  1. ^ Hopf, H.; Rinow, W. (1931). "Ueber den Begriff der vollständigen differentialgeometrischen Fläche". Commentarii Mathematici Helvetici. 3 (1): 209-225. doi:10.1007/BF01601813.
  2. ^ Atkin, C. J. (1975), "The Hopf-Rinow theorem is false in infinite dimensions" (PDF), The Bulletin of the London Mathematical Society, 7 (3): 261-266, doi:10.1112/blms/7.3.261, MR 0400283.
  3. ^ O'Neill, Barrett (1983), Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, p. 193, ISBN 9780080570570.

References


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Hopf%E2%80%93Rinow_theorem
 



 



 
Music Scenes