Local invariants and geometry of the sub-Laplacian on H-type foliations

Abstract

$H$-type foliations $(M,H,g_H)$ are studied in the framework of sub-Riemannian geometry with bracket generating distribution defined as the bundle transversal to the fibers. Equipping $M$ with the Bott connection we consider the scalar horizontal curvature ${\kappa }_H$ as well as a new local invariant ${\tau }_V$ induced from the vertical distribution. We extend recent results on the small-time asymptotics of the sub-Riemannian heat kernel on quaternion-contact (qc-)manifolds due to A. Laaroussi and we express the second heat invariant in sub-Riemannian geometry as a linear combination of ${\kappa }_H$ and ${\tau }_V$. The use of an analog to normal coordinates in Riemannian geometry that are well-adapted to the geometric structure of $H$-type foliations allows us to consider the pull-back of Korányi balls to $M$. We explicitly obtain the first three terms in the asymptotic expansion of their Popp volume for small radii. Finally, we address the question of when $M$ is locally isometric as a sub-Riemannian manifold to its $H$-type tangent group.