On Jacobi fields and canonical connection in sub-Riemannian geometry

TL;DR

This paper interprets the coefficients of the Jacobi equation in sub-Riemannian geometry as curvature of a canonical nonlinear Ehresmann connection, providing new insights into geometric invariants.

Contribution

It introduces a novel interpretation of Jacobi equation coefficients as curvature of a canonical nonlinear connection in sub-Riemannian geometry.

Findings

Coefficients of Jacobi equation are curvature invariants.

The canonical connection is nonlinear and has specific properties.

Provides a geometric framework linking Jacobi fields and connections.

Abstract

In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first introduced in [Zelenko-Li]. We show why this connection is naturally nonlinear, and we discuss some of its properties.