A Natural Connection on (2,3) Sub-Riemannian Manifolds

TL;DR

This paper introduces a natural analogue of the Levi-Civita connection for (2,3) sub-Riemannian manifolds modeled on the Heisenberg group, establishing its uniqueness and geometric properties.

Contribution

It constructs a unique connection for sub-Riemannian manifolds that parallels the Levi-Civita connection in Riemannian geometry, with demonstrated geometric properties.

Findings

The connection is unique with specified properties.

It exhibits key geometric features analogous to Levi-Civita.

The connection provides a new tool for sub-Riemannian geometry.

Abstract

We build an analogue for the Levi-Civita connection on Riemannian manifolds for sub-Riemannian manfiolds modeled on the Heisenberg group. We demonstrate some geometric properties of this connection to justify our choice and show that this connection is unique in having these properties.