Isometries of complemented subRiemannian manifolds

TL;DR

This paper proves that the isometry group of complemented sub-Riemannian manifolds is a Lie group, provides dimension estimates, and generalizes classical Riemannian results to the sub-Riemannian setting with applications to specific groups.

Contribution

It establishes the Lie group structure of isometries, derives dimension bounds, and extends classical Bochner and Berger theorems to complemented sub-Riemannian manifolds.

Findings

Isometry groups form Lie groups with dimension estimates

Derived a Bochner formula for Killing fields in sub-Riemannian geometry

Computed isometry groups for examples like SO(n), SL(n), and rototranslation group

Abstract

We show that the group of smooth isometries of a complemented sub-Riemannian manifold form a Lie group and establish dimension estimates based on the torsion of the canonical connection. We explore the interaction of curvature and the structure of isometries and Killing fields and derive a Bochner formula for Killing fields. Sub-Riemannian generalizations of classical results of Bochner and Berger are established. We also apply our theory to common sub-categories of complemented sub-Riemannian geometries and show how to compute the isometry groups for several examples, including SO(n), SL(n) and the rototranslation group.