Triple Products of Eigenfunctions and Spectral Geometry

Using elementary techniques from Geometric Analysis, Partial Differential Equations, and Abelian $C^{\ast }$ Algebras, we uncover a novel, yet familiar, global geometric invariant – namely the indexed set of integrals of triple products of eigenfunctions of the Laplace-Beltrami operator, to precisely characterize which isospectral closed Riemannian manifolds are isometric.