Constant Curvature Models in Sub-Riemannian Geometry

TL;DR

This paper explores constant curvature models in sub-Riemannian geometry, focusing on their invariants and cohomological structures, especially within parabolic geometries, and illustrates these models with specific examples.

Contribution

It provides a cohomological framework for understanding curvature invariants in constant curvature sub-Riemannian geometries, linking them to Cartan connections and parabolic structures.

Findings

Cohomological description of curvature invariants

Unique Cartan connections for constant curvature models

Illustrative examples of specific sub-Riemannian geometries

Abstract

Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description of the structure of these curvature invariants in the cases where the background structure is one of the parabolic geometries. As an illustration, constant curvature models are discussed for certain sub-Riemannian geometries.