Curvature and the equivalence problem in sub-Riemannian geometry

TL;DR

This paper introduces methods to determine when sub-Riemannian manifolds are isometric by analyzing their curvature, providing concrete examples including Engel, contact, and Cartan manifolds, with some original research on Engel manifolds.

Contribution

It develops the description of canonical grading and connections for sub-Riemannian manifolds with constant symbol, aiding in the equivalence problem, and presents original work on Engel manifolds.

Findings

Canonical grading and connection characterize isometry classes.

Explicit examples of Engel, contact, and Cartan manifolds.

Original research on Engel (2,3,4)-manifolds with minimal isometry groups.

Abstract

These notes give an introduction to the equivalence problem of sub-Riemannian manifolds. We first introduce preliminaries in terms of connections, frame bundles and sub-Riemannian geometry. Then we arrive to the main aim of these notes, which is to give the description of the canonical grading and connection existing on sub-Riemann manifolds with constant symbol. These structures are exactly what is needed in order to determine if two manifolds are isometric. We give three concrete examples, which are Engel (2,3,4)-manifolds, contact manifolds and Cartan (2,3,5)-manifolds. These notes are an edited version of a lecture series given at the \href{https://conference.math.muni.cz/srni/}{42nd Winter school: Geometry and Physics}, Snr\'i, Check Republic, mostly based on other earlier work. However, the work on Engel (2,3,4)-manifolds is original research, and illustrate the important special case were our model has the minimal set of isometries.