Myers's theorem
The Myers theorem, also known as the Bonnet–Myers theorem, is a classical and well-known theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941.
Let (M,g) be a complete and smooth Riemannian manifold of dimension n. If k is a positive number with Ricg ≥ (n-1)k, then any curve of length greater than π/√k can be shortened.
Precisely, this says that if γ : [a,b]→M is a smooth path of length greater than π/√k, then there exists ε > 0 and for each s ∈ (-ε,ε) a smooth path γs : [a,b]→M with γs(a)=γ(a) and with γs(b)=γ(b), with γ0 = γ, and such that the associated map (-ε,ε)×[a,b]→M is smooth, and such that the length of γs is less than that of γ for all s ∈ (0,ε). In a topological language, this says that there is a smooth homotopy of γ with fixed endpoints and which decreases length.
An earlier result (from 1855), due to Ossian Bonnet, has the same conclusion but under the weaker assumption that the sectional curvatures is bounded below by k.
Corollaries
The conclusion of the theorem says, in particular, that the diameter of (M,g) is finite. The Hopf-Rinow theorem implies that M must be compact, as it is closed and bounded.
As a very particular case, this shows that any complete and noncompact smooth Riemannian manifold which is Einstein must have nonpositive Einstein constant.
Consider the smooth universal covering map π : N→M. One may consider the Riemannian metric π*g on N. Since π is a local diffeomorphism, Myers' theorem applies to the Riemannian manifold (N,π*g) and hence N is compact. This implies that the fundamental group of M is finite.
Cheng's diameter rigidity theorem
The conclusion of Myers' theorem says that for any p and q in M, one has dg(p,q) ≤ π/√k. In 1975, Shiu-Yuen Cheng proved:
Let (M, g) be a complete and smooth Riemannian manifold of dimension n. If k is a positive number with Ricg ≥ (n-1)k, and if there exists p and q in M with dg(p,q) = π/√k, then (M,g) is simply-connected and has constant sectional curvature k.
References
- Ambrose, W. A theorem of Myers. Duke Math. J. 24 (1957), 345–348.
- Cheng, Shiu Yuen (1975), "Eigenvalue comparison theorems and its geometric applications", Mathematische Zeitschrift, 143 (3): 289–297, 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): 401–404, doi:10.1215/S0012-7094-41-00832-3