Local Invariants and Geometry of the sub-Laplacian on H-type Foliations

TL;DR

This paper investigates the geometry of $H$-type foliations in sub-Riemannian spaces, introducing new invariants and extending heat kernel asymptotics, leading to explicit volume expansion and isometry characterizations.

Contribution

It introduces a new local invariant $ au_{ u}$, extends heat kernel asymptotics to $H$-type foliations, and provides explicit volume expansion and criteria for local isometry to tangent groups.

Findings

Expressed the second heat invariant as a combination of $ ext{scalar curvature}$ and $ au_{ u}$.

Derived the first three terms in the asymptotic expansion of Popp volume.

Characterized conditions for local isometry to $H$-type tangent groups.

Abstract

$H$-type foliations $(\mathbb{M},\mathcal{H},g_{\mathcal{H}})$ are studied in the framework of sub-Riemannian geometry with bracket generating distribution defined as the bundle transversal to the fibers. Equipping $\mathbb{M}$ with the Bott connection we consider the scalar horizontal curvature $\kappa_{\mathcal{H}}$ as well as a new local invariant $\tau_{\mathcal{V}}$ induced from the vertical distribution. We extend recent results on the small-time asymptotics of the sub-Riemannanian 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 $\kappa_{\mathcal{H}}$ and $\tau_{\mathcal{V}}$. 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 Kor\'{a}nyi balls to $\mathbb{M}$. 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 $\mathbb{M}$ is locally isometric as a sub-Riemannian manifold to its $H$-type tangent group.