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$H$-type foliations $(\mathbb{M},\mathcal{H},g_{\mathcal{H}})$ are studied inthe framework of sub-Riemannian geometry with bracket generating distributiondefined as the bundle transversal to the fibers. Equipping $\mathbb{M}$ withthe 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 thesmall-time asymptotics of the sub-Riemannanian heat kernel onquaternion-contact (qc-)manifolds due to A. Laaroussi and we express the secondheat invariant in sub-Riemannian geometry as a linear combination of$\kappa_{\mathcal{H}}$ and $\tau_{\mathcal{V}}$. The use of an analog to normalcoordinates in Riemannian geometry that are well-adapted to the geometricstructure of $H$-type foliations allows us to consider the pull-back ofKor\'{a}nyi balls to $\mathbb{M}$. We explicitly obtain the first three termsin the asymptotic expansion of their Popp volume for small radii. Finally, weaddress the question of when $\mathbb{M}$ is locally isometric as asub-Riemannian manifold to its $H$-type tangent group.

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