Abstract: This dissertation explores the extent to which lengths of closed geodesics on a Riemannian manifold determine the underlying metric. In the setting of closed manifolds of negative sectional curvature, it is known in certain cases---and conjectured to be true in general---that in order to determine a Riemannian metric on a given manifold up to isometry, it suffices to measure the lengths of all closed geodesics (as a function of their free homotopy classes). This phenomenon is known as marked length spectrum rigidity.

In this dissertation, we prove quantitative versions of some previously known marked length spectrum rigidity results. We consider pairs of Riemannian metrics where the lengths of closed geodesics agree---only approximately---on a finite set of free homotopy classes of curves. We prove the two metrics are "approximately isometric", meaning bi-Lipschitz equivalent with constant close to 1. We obtain explicit estimates for this Lipschitz constant in terms of the measurement error and the length of the longest geodesic in the finite set. Our estimates depend only on concrete geometric information about the given metrics, such as the dimension, sectional curvature bounds, and injectivity radii.

Hybrid Event Zoom Link: https://umich.zoom.us/j/98478787716?pwd=VUp2SE16TVZMTnladTVmdjNpMTRMdz09