Inferring geometric structure from eigenfunction triple products

This article presents two converses to well-known observations regarding products of eigenfunctions of the Laplacian on a closed Riemannian manifold $(M,g)$. First, if $(M,g)$ is real-analytic, the coefficients in the eigenfunction expansion of a product of eigenfunctions decay exponentially. We show that such exponential decay in fact implies that $g$ is real-analytic. Second, it is clear that if $(M,g)$ is flat, there exists an eigenbasis for which the product of any two of its elements lies in the span of at most $N = N(M,g)$ elements. We prove that any manifold with such an eigenbasis must be flat. We then construct metrics on $\mathbb{S}^7$ that share an eigenbasis with the standard metric yet are not locally homogeneous, from which we deduce that homogeneity cannot be characterized solely in terms of the multiplicative structure of eigenfunctions.