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:
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
A corollary of this is that if the sectional curvature is bounded below by —which implies by the boxed equation above that for all , because we can choose the orthonormal basis above such that are orthogonal to —then
Also, the finite diameter implies compactness since is complete as a metric space.
The fundamental group
Theorem 2 Let be a compact Riemannian manifold with positive Ricci curvature; then its fundamental group is finite.
I think this is especially interesting because it relates the Ricci curvature of the manifold, which is defined heavily in terms of the Riemannian metric and not in terms of itself, with the bare-bones topology of expressed in the fundamental group.
The proof, however, is fairly straightforward from what we have already done. It will be sufficient to show that the universal covering space of is compact, because if is the covering map, then for is discrete and closed, hence finite. So is finitely-sheeted over , which implies the theorem.
I will now show is compact.
Now it is a basic fact that any covering space of a smooth manifold can be made into a smooth manifold (this is also true with “smooth” replaced by real-analytic, complex, etc.). Moreover, we can pull back the Riemannian metric on via to get on . Since is locally an isometry, it preserves curvature; in particular, has bounded-below Ricci curvature because does.
Moreover, is a complete Riemannian manifold. Indeed, more generally, any covering space of a complete Riemannian manifold—with the pulled-back Riemannian metric–is complete. The reason is that if is the covering map, a curve is a geodesic iff is one; this too follows from the local isometry property of . Therefore, if we start at a point and start a geodesic from , we can project it to via , extend it to a geodesic on on by completeness, and use the covering space property to lift it upward to (uniquely since the starting point is fixed) to get a geodesic in . So geodesics in are infinitely extendable, which implies completeness.
Now the Myers theorem applied to shows compactness and establishes the second theorem.