I'm studying the Laplacian on (compact) Riemannian manifolds, and it turns out that if the Laplacian operators of two such spaces share their spectrum (the spaces are then called isospectral), then the spaces themselves must share some geometric properties. The most spectacular I've come across is that if two Riemannian manifolds are isospectral, then they share their sequence of lengths of closed geodesics. What other properties do they share? I'd really appreciate references on these, so I could look up the proofs.
So far I've looked at Buser's textbook "Geometry and Spectra of Compact Riemann Surfaces", and some notes by Yaiza Canzani for a course at Harvard. Between them they mention the closed geodesic lengths, as well as shared volume, dimension, and total Ricci curvature (so for surfaces, the same Euler characteristic, meaning that they are diffeomorphic assuming compactness).