Abstract



To each secondorder 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 odddegree characteristic 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 secondorder differential invariants factor through the curvature mapping; a structure is thus established for KCC theory.  
International

Si 
JCR

Si 
Title

Differential Geometry and its applications 
ISBN

09262245 
Impact factor JCR

0,65 
Impact info


Volume

31 


Journal number


From page

437 
To page

462 
Month

SIN MES 
Ranking

Mathematics 