Geometric Spectra

Given a Riemannian manifold M, one can try to study the geometry of the M by studying the totally geodesic submanifolds of M. The 1-dimensional case yields the associated geodesic length spectrum, an important invariant with ties to spectral geometry. In this talk, I will give a brief survey of some results on the geodesic length spectrum and generalizations of these results to the 2-dimensional case of totally geodesic surfaces. The talk will focus almost exclusively on compact locally symmetric manifolds and more specifically real hyperbolic 3-manifolds. This work is joint with Alan Reid from the University of Texas.