Myers's theorem
The Myers theorem, also known as the Bonnet–Myers theorem, is a classical theorem in Riemannian geometry. The strong form was proven by Sumner Byron Myers. The theorem states that if Ricci curvature of an n-dimensional complete Riemannian manifold M is bounded below by (n − 1)k > 0, then its diameter is at most π/√k. In particular, this shows that any such M is necessarily compact. A weaker result, due to Ossian Bonnet, has the same conclusion but under the stronger assumption that the sectional curvatures is bounded below by k.
Moreover, if the diameter is equal to π/√k, then the manifold is isometric to a sphere of a constant sectional curvature k. This rigidity result is due to Cheng (1975), and is often known as Cheng's theorem.
This result also holds for the universal cover of such a Riemannian manifold, in particular both M and its cover are compact, so the cover is finite-sheeted and M has finite fundamental group.
See also
References
- Cheng, Shiu Yuen (1975), "Eigenvalue comparison theorems and its geometric applications", Mathematische Zeitschrift, 143 (3): 289&ndash, 297, doi:10.1007/BF01214381, ISSN 0025-5874, MR 0378001
- do Carmo, M. P. (1992), Riemannian Geometry, Boston, Mass.: Birkhäuser, ISBN 0-8176-3490-8
- Myers, S. B. (1941), "Riemannian manifolds with positive mean curvature", Duke Mathematical Journal, 8 (2): 401&ndash, 404, doi:10.1215/S0012-7094-41-00832-3