Myers's theorem

The conclusion of the theorem says, in particular, that the diameter of (M, g) is finite. Therefore M must be compact, as a closed (and hence compact) ball of finite radius in any tangent space is carried onto all of M by the exponential map. As a very particular case, this shows that any complete and noncompact smooth Einstein manifold must have nonpositive Einstein constant. Since M is connected, there exists the smooth universal covering map \pi : N \to M. One may consider the pull-back metric \pi^*g on N. Since \pi is a local isometry, Myers' theorem applies to the Riemannian manifold (N,\pi^*g) and hence N is compact and the covering map is finite. 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, q \in M, one has d_g(p,q)\leq\frac{\pi}{\sqrt{k}}. In 1975, Shiu-Yuen Cheng proved: {{quote|Let (M, g) be a complete and smooth Riemannian manifold of dimension . If is a positive number with , and if there exists and in with d_g(p,q)=\frac{\pi}{\sqrt{k}}, then (M,g) is simply-connected and has constant sectional curvature .}} == See also ==