Trivializable and quaternionic subriemannian structure on $\mathbb{S}^7$ and subelliptic heat kernel

TL;DR

This paper compares two distinct subriemannian structures on the 7-sphere, analyzing their geometric and spectral properties, and demonstrating they are not locally isometric or isospectral.

Contribution

It introduces and contrasts trivializable and quaternionic subriemannian structures on $ ext{S}^7$, providing explicit measures, sublaplacians, and spectral analysis.

Findings

Both structures are not locally isometric.

Neither structure admits a global nowhere vanishing smooth section.

The structures are not isospectral in the subriemannian sense.

Abstract

On the seven dimensional Euclidean sphere $\mathbb{S}^7$ we compare two subriemannian structures with regards to various geometric and analytical properties. The first structure is called trivializable and the underlying distribution $\mathcal{H}_T$ is induced by a Clifford module structure of $\mathbb{R}^8$. More precisely, $\mathcal{H}_T$ is rank $4$, bracket generating of step two and generated by globally defined vector fields. The distribution $\mathcal{H}_{Q}$ of the second structure is of rank 4 and step two as well and obtained as the horizontal distribution in the quaternionic Hopf fibration $\mathbb{S}^3\hookrightarrow\mathbb{S}^7\rightarrow\mathbb{S}^4$. Answering a question in arXiv:0901.1406 we first show that $\mathcal{H}_{Q}$ does not admit a global nowhere vanishing smooth section. In both cases we determine the Popp measures, the intrinsic sublaplacians $\Delta_{sub}^T$ and $\Delta_{sub}^{Q}$ and the nilpotent approximations. We conclude that both subriemannian structures are not locally isometric and we discuss properties of the isometry group. By determining the first heat invariant of the sublaplacians it is shown that both structures are also not isospectral in the subriemannian sense.