Isospectral closed Riemannian manifolds which are not locally isometric

Abstract

Two compact Riemannian manifolds are said to be isospectral if the associated Laplace-Beltrami operators have the same eigenvalue spectrum. Milnor [18] constructed the first pair of isospectral, nonisometric manifolds, a pair of 16-dimensional flat tori. Many new examples and also techniques for constructing examples have appeared in the past decade; see for example [2], [3], [4], [6], [10], [11], [12], [13], [22] and [25] or the surveys [1], [2], [5] and [8]. However, in all these examples, the isospectral manifolds are locally isometric; in particular, in all the examples of isospectral closed Riemannian manifolds, the manifolds have a common Riemannian covering. Recently, Zoltan Szabo [24] constructed the first examples of isospectral Riemannian manifolds (with boundary) which are not locally isometric. The manifolds are geodesic balls in different harmonic manifolds of nonpositive curvature. These manifolds are first introduced in [23].