Geometric structures associated with the Chern connection attached to a SODE

Abstract

To each second-order ordinary differential equation $σ$ on a smooth manifold $M$ a $G$-structure $P^σ$ on $J^1(\mathbb{R},M)$ is associated and the Chern connection $\nabla ^σ$ attached to $σ$ is proved to be reducible to $P^σ$; in fact, $P^σ$ coincides generically with the holonomy bundle of $\nabla ^σ$. 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 $P^σ$ with prescribed torsion tensor field. The properties of the curvature tensor field of $\nabla ^σ$ in relationship to the existence of special coordinate systems for $σ$ are studied. Moreover, all the odd-degree characterictic classes on $P^σ$ are seen to be exact and the usual characteristic classes induced by $\nabla ^σ$ 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 $\nabla ^σ$ 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.