Myers Theorem

Myers Theorem is a classical theorem in Riemannian geometry .

Content

Wording

If the curvature of Ricci complete $n$ ${\ displaystyle n}$ -dimensional Riemannian manifold $M$ ${\ displaystyle M}$ bounded below by a positive value $(n-1)k$ ${\ displaystyle (n-1) k}$ for some $k$ ${\ displaystyle k}$ , then its diameter does not exceed $\pi /{\sqrt {k}}$ ${\ displaystyle \ pi / {\ sqrt {k}}}$ . Moreover, if the diameter is $\pi /{\sqrt {k}}$ ${\ displaystyle \ pi / {\ sqrt {k}}}$ , then the manifold itself is isometric to the sphere of constant sectional curvature $k$ ${\ displaystyle k}$ .

Consequences

This result remains valid for the universal covering of such a Riemannian manifold. $M$ ${\ displaystyle M}$ . In particular, the universal cover $M$ ${\ displaystyle M}$ finitely leafed and means fundamental group $\pi _{1}M$ ${\ displaystyle \ pi _ {1} M}$ finite.

History

A similar result for sectional curvature was proved earlier by Bonnet .

The theorem is proved by Myers . ^[1] The case of equality in the theorem was proved by Cheng in 1975. ^[2]

Notes

↑ Myers, SB (1941), " Riemannian manifolds with positive mean curvature ", Duke Mathematical Journal T. 8 (2): 401–404 , DOI 10.1215 / S0012-7094-41-00832-3
↑ Cheng, Shiu Yuen (1975), " Eigenvalue comparison theorems and its geometric applications ", Mathematische Zeitschrift T. 143 (3): 289–297, ISSN 0025-5874 , DOI 10.1007 / BF01214381

[1] Myers, SB (1941), " Riemannian manifolds with positive mean curvature ", Duke Mathematical Journal T. 8 (2): 401–404 , DOI 10.1215 / S0012-7094-41-00832-3

[2] Cheng, Shiu Yuen (1975), " Eigenvalue comparison theorems and its geometric applications ", Mathematische Zeitschrift T. 143 (3): 289–297, ISSN 0025-5874 , DOI 10.1007 / BF01214381