On a (compact) Riemannian manifold there is a natural differential operator on functions, the Laplacian.  The eigenvalues of this operator are important invariants of the manifold, and there are many interesting results which relate the eigenvalues to other geometric quantities (volume, diameter, bounds on curvature, and so on).  In the project you could look at some of these inequalities, and their proof and applications, using a wide variety of methods.