Geometric structures associated with the Chern connection attached to a SODE

TL;DR

This paper explores the geometric structures linked to the Chern connection derived from second-order differential equations on manifolds, revealing their properties, invariants, and relation to characteristic classes.

Contribution

It introduces a $G$-structure associated with each SODE, characterizes the Chern connection, and connects curvature and invariants to the geometry of differential equations.

Findings

Chern connection reducible to a specific $G$-structure

All odd-degree characteristic classes are exact

Maximal automorphism group is the group of $p$-vertical automorphisms

Abstract

To each second-order ordinary differential equation $\sigma $ on a smooth manifold $M$ a $G$-structure $P^\sigma $ on $J^1(\mathbb{R},M)$ is associated and the Chern connection $\nabla ^\sigma $ attached to $\sigma $ is proved to be reducible to $P^\sigma $; in fact, $P^\sigma $ coincides generically with the holonomy bundle of $\nabla ^\sigma $. The cases of unimodular and orthogonal holonomy are also dealt with. Two characterizations of the Chern connection are given: The first one in terms of the corresponding covariant derivative and the second one as the only principal connection on $P^\sigma $ with prescribed torsion tensor field. The properties of the curvature tensor field of $\nabla ^\sigma $ in relationship to the existence of special coordinate systems for $\sigma $ are studied. Moreover, all the odd-degree characterictic classes on $P^\sigma $ are seen to be exact and the usual characteristic classes induced by $\nabla ^\sigma $ determine the Chern classes of $M$. The maximal group of automorphisms of the projection $p\colon \mathbb{R}\times M\to \mathbb{R}$ with respect to which $\nabla ^\sigma $ has a functorial behaviour, is proved to be the group of $p$-vertical automorphisms. The notion of a differential invariant under such a group is defined and stated that second-order differential invariants factor through the curvature mapping; a structure is thus established for KCC theory.