Myers's theorem

Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the following:

Let $(M, g)$ be a complete and connected Riemannian manifold of dimension $n$ whose Ricci curvature satisfies for some fixed positive real number $r$ the inequality $\operatorname{Ric}_{p}(v)\geq (n-1)\frac{1}{r^2}$ for every $p\in M$ and $v\in T_{p}M$ of unit length. Then any two points of M can be joined by a geodesic segment of length at most $\pi r$.

In the special case of surfaces, this result was proved by Ossian Bonnet in 1855. For a surface, the Gauss, sectional, and Ricci curvatures are all the same, but Bonnet's proof easily generalizes to higher dimensions if one assumes a positive lower bound on the  sectional curvature. Myers' key contribution was therefore to show that a Ricci lower bound is all that is needed to reach the same conclusion.

Corollaries
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 Riemannian manifold which is Einstein 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 $π^{*}g$ on $$N.$$ Since $$\pi$$ is a local isometry, Myers' theorem applies to the Riemannian manifold $(N,π^{*}g)$ 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 $d_{g}(p,q) ≤ π/\sqrt{k}$. 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 $Ric^{g} ≥ (n-1)k$, and if there exists $p$ and $q$ in $M$ with $d_{g}(p,q) = π/\sqrt{k}$, then $(M,g)$ is simply-connected and has constant sectional curvature $k$."