Spectral Geometry on Nilmanifolds

Two Riemannian manifolds are said to be isospectral if the associated Laplace-Beltrami operators have the same spectrum. Riemannian nilmanifolds have provided a rich source of examples of isospectral manifolds, exhibiting a wide variety of different phenomena. In particular, there exist continuous families of isospectral, nonisometric nil-manifolds, isospectral nilmanifolds for which the Laplacians acting on one-forms are not isospectral, and isospectral nilmanifolds that are not even locally isometric. This article reviews three different methods for constructing isospectral nilmanifolds and examines the geometry of resulting examples. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.