Local Equivalence Problem for Sub-Riemannian Structures

TL;DR

This paper addresses the local equivalence problem for sub-Riemannian structures on odd-dimensional manifolds, establishing criteria based on canonical linear connections and exploring special cases and dimensional constraints.

Contribution

It provides a solution to the local equivalence problem for sub-Riemannian structures and relates these to generalized Tanaka-Webster connections in specific cases.

Findings

Two sub-Riemannian structures are locally equivalent iff their canonical connections are equivalent.

In dimension 3, these connections match the generalized Tanaka-Webster connection.

In dimensions greater than 5, some contact metric manifolds may not correspond to any given sub-Riemannian structure.

Abstract

We solve the local equivalence problem for sub-Riemannian structures on (2n + 1)-dimensional manifolds. We show that two sub-Riemannian structures are locally equivalent if and only if? their corresponding canonical linear connections are equivalent. When n = 1, these connections coincide with the generalized Tanaka-Webster connection of the corresponding contact metric structure. We show that in dimension > 5, there may not be any contact metric manifolds associated with a given sub-Riemannian structure.