Algebraic topology & Riemannian geometry project idea?

I'm not sure if this is what you have in mind, but here are a couple of theorems that concern the interaction between topology and Riemannian geometry.

• Gauss-Bonnet theorem, and its generalization, sometimes called the Chern-Gauss-Bonnet theorem. It relates the curvature of a Riemannian metric on a manifold to its topology (Euler characteristic).
• Bonnet-Myers theorem, which implies that a compact Riemannian manifold with positive Ricci curvature has finite fundamental group.

Jurgen Jost's Riemannian Geometry and Geometric Analysis has a chapter "A Short Survey on Curvature and Topology" summarizing the two theorems above and a few other important theorems in the same spirit. Jost paraphrases a fundamental question of Hopf: "To what extent does the existence of a Riemannian metric with particular curvature properties restrict the topology of the underlying differentiable manifold?" The answers to this question (in my opinion) are very interesting!

This is a fun question for me, since I studied precisely a connection between the two.

In Riemannian geometry, a natural question is: Are there closed geodesics in my manifold? How many of them? What can I say about them?

On the other hand, when you deal with Morse theory, you want to relate the topology of your manifold with the number of critical points of a given Morse function.

The caveat is: closed geodesics are critical points of a differentiable function (though not a Morse function... but still good enough for Morse theory in some spaces): the energy functional. However, you have to consider the Free Loop Space as your manifold of interest, which is usually denoted by $\Lambda M$ in the literature. You then have the job of computing the homology of $\Lambda M$ in order to obtain information about the closed geodesics (this has been used, for example, to prove that there are at least two closed geodesics in the sphere with a Finsler metric, though I did not study this part of the theory).

Therefore, we can connect Morse theory (which uses instruments from algebraic topology) with Riemannian geometry this way. Note also that, in the way, a whole list of mathematical subjects can be approached, depending on how much deep in details you want to go. For example, considerable Analysis happens, since $\Lambda M$ is the map of $H^1$ functions from the circle to the manifold. And it must be so, because we want a model on a complete space, which is Hilbert and has a nice enough way of taking derivatives. There also is functional analysis, since we are dealing with Hilbert spaces etc.