Spectral analysis of classes of subriemannian manifolds

Abstract

In this thesis we study classes of subriemannian manifolds and their sublaplacians. We determine the spectrum of the intrinsic sublaplacian on pseudo H-type nilmanifolds. Moreover, we construct an arbitrary number of isospectral but pairwise non-homeomorphic pseudo H-type nilmanifolds with respect to the sublaplacian.

Furthermore, with the help of the representation theory of step $2$ nilpotent Lie groups, we show that the spectrum of the intrinsic sublaplacian on nilmanifolds whose covering space is a step $2$ Carnot group can be completely expressed in terms of the underlying metric Lie algebra data. In the case of H-type nilmanifolds whose covering space has odd dimensional center, we prove a Poisson summation formula that relates the spectrum of the intrinsic sublaplacian to the lengths of closed subriemannian geodesics. Based on the nilpotent approximation of subriemannian manifolds, we compare two subriemannian structures of step $2$ and rank $4$ on the Euclidean sphere $\mathbb{S}^7$ with regards to various geometric and analytical properties. We show that both subriemannian structures are neither locally isometric nor isospectral with respect to the subaplacian. Finally, we study the heat kernel associated to the intrinsic sublaplacian on a quaternionic contact manifold considered as a subriemannian manifold. More precisely, we explicitly compute the first two coefficients appearing in the small time asymptotic expansion of the heat kernel on the diagonal. We show that the second coefficient coincides with the quaternionic contact scalar curvature up to a (universal) constant multiple.