Hearing the shape of a drum by knocking around

Abstract We study a variation of Kac's question, “Can one hear the shape of a drum?” if we allow ourselves access to some additional information. In particular, we allow ourselves to “hear” the local Weyl counting function at each point on the manifold and ask if this is enough to uniquely recover the Riemannian metric. This is physically equivalent to asking whether one can determine the shape of a drum if one is allowed to knock at any place on the drum. We show that the answer to this question is “yes” provided the Laplace–Beltrami spectrum of the drum is simple. We also provide a counterexample illustrating why this hypothesis is necessary. As a corollary, we give a short proof that manifolds with simple Laplace–Beltrami spectra have finite isometry groups.