Geometric structures associated with the Chern connection attached to a SODE

To each second-order ordinary differential equation σ on a smooth manifold M a G-structure Pσ on J1(R,M) is associated and the Chern connection ∇σ attached to σ is proved to be reducible to Pσ; in fact, Pσ coincides generically with the holonomy bundle of ∇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 Pσ with prescribed torsion tensor field. The properties of the curvature tensor field of ∇σ in relationship to the existence of special coordinate systems for σ are studied. Moreover, all the odd-degree characteristic classes on Pσ are seen to be exact and the usual characteristic classes induced by ∇σ determine the Chern classes of M. The maximal group of automorphisms of the projection p:R×M→R with respect to which ∇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.