Spectral geometry of Subriemannian structures on nilmanifolds (Symmetry and Singularity of Geometric Structures and Differential Equations)

Abstract

This paper is a survey on recent results on spectral geometry for the Sublaplacian on certain compact nilmanifolds. We determine the spectrum of the Sublaplacian on compact quotients of 2-step Carnot groups by a lattice and focus on a class of generalized H-type groups. We prove a Poisson summation formula relating the spectrum of the Sublaplacian and lengths of closed Subriemannian geodesics.