Near Isospectrality and Spectral Rigidity for Compact Locally Symmetric Manifolds

The inverse spectral problem asks to what extent the Laplace--Beltrami spectrum determines the geometry of a Riemannian manifold. We study a natural weakening, called \emph{near isospectrality}, in which the spectra of two compact manifolds agree outside a finite set, counted with multiplicity. We prove that for compact quotients of a fixed simply connected symmetric space of nonpositive sectional curvature, near isospectrality already forces full isospectrality. We then extend this rigidity to a broad collection of compact quotients of irreducible symmetric spaces of noncompact type. In this larger setting, near isospectrality determines enough heat invariants to identify the universal cover within the class under consideration, and the fixed-cover rigidity result then implies full isospectrality. Thus, within the class studied here, eventual agreement of the Laplace spectrum already forces complete spectral agreement.