*I edited this post to fix some sign issues. (11/29)*

I now want to discuss a result of Myers, which I can summarize as follows:

*If is a complete Riemannian manifold with positive, bounded-below curvature, then is compact.*

This is a very loose summary—Myers’ theorem actually gives a lower bound for the diameter of . Moreover, I haven’t explained what “bounded below curvature” actually means. To say that the sectional curvature is bounded below is sufficient, but we can do better.

I will now outline how the proof works.

The first thing to notice is that any two points can be joined by a length-minimizing geodesic , by the Hopf-Rinow theorems. In particular, if we can show that every sufficiently long geodesic (of length , say) **doesn’t** minimize length, then is necessarily of diameter at most .

All the same, the length function as a map is not so easy to work with; the energy integral is much more convenient. Moreover, we know that if minimizes length and is a geodesic, it also minimizes the energy integral.

If is a geodesic that minimizes the energy integral (among curves with fixed endpoints), then in particular we can consider a variation of , and consider the function ; this necessarily has a minimum at . It follows that . If we apply the second variation formula, we find something involving the curvature tensor that looks a lot like sectional curvature.

Before turning to the details, I will now define the refinement of sectional curvature we can use:

**Ricci curvature **

Given a Riemannian manifold with curvature tensor and , we can define a linear map ,

that depends on . The trace of this linear map is defined to be the **Ricci tensor** . This is an invariant definition, so we do not have to do any checking of transformation laws.

A convenient way to express this is the following: If is an orthonormal basis for , then by linear algebra and skew-symmetry

The Ricci curvature has many uses. Considered as a tensor (by the functorial isomorphism for any real vector space and the isomorphism induced by the Riemannian metric), its trace yields the **scalar curvature**, which is just a real-valued function on a Riemannian manifold. It is also used in defining the Ricci flow, which led to the recent solution of the Poincaré conjecture. I may talk about these more advanced topics (much) later if I end up learning about them–I am finding this an interesting field, and may wish to pursue geometry further.

**Statement of Myers’ theorem **

**Theorem 1** *Let be a complete Riemannian manifold whose Ricci tensor satisfies *

*for all . Then*