  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.
