Sparse spectrally rigid sets for negatively curved manifolds

Abstract

Suppose that $(M,\mathfrak{g})$ is a compact Riemannian manifold with strictly negative sectional curvatures. A subset of conjugacy classes $E \subset \text{conj}(\pi_1(M))$ is called spectrally rigid if when two negatively curved Riemannian metrics $\mathfrak{g}_1, \mathfrak{g}_2$ on $M$ have the same marked length spectrum on $E$, then their marked length spectra coincide everywhere. In this work we show that there are arbitrarily sparse spectrally rigid sets and that they exist, in some sense, in every direction in $\pi_1(M)$.