Given a fibered manifold , a 2-connection on means a section . The authors determine all first order natural operators transforming a 2-connection on and a classical linear connection on into a connection on . (The proof implies that there is no first order natural operator transforming 2-connections on into connections on .) Using this result, the authors deduce several properties of characterizable connections on .
The main goal of the present work is a generalization of the ideas, constructions and results from the first and second-order situation, studied in [63], [64] to that of an arbitrary finite-order one. Moreover, the investigation extends the ideas of [65] from the one-dimensional base X corresponding to O.D.E.
The (infinitesimal) symmetries of first and second-order partial differential equations represented by connections on fibered manifolds are studied within the framework of certain “strong horizontal“ structures closely related to the equations in question. The classification and global description of the symmetries is presented by means of some natural compatible structures, eġḃy vertical prolongations of connections.
The homogeneity properties of two different families of geometric objects playing a crutial role in the non-autonomous first-order dynamics - semisprays and dynamical connections on - are studied. A natural correspondence between sprays and a special class of homogeneous connections is presented.
The geometry of second-order systems of ordinary differential equations represented by -connections on the trivial bundle is studied. The formalism used, being completely utilizable within the framework of more general situations (partial equations), turns out to be of interest in confrontation with a traditional approach (semisprays), moreover, it amounts to certain new ideas and results. The paper is aimed at discussion on the interrelations between all types of connections having to do with...
We present a generalization of the concept of semiholonomic jets within the framework of higher order prolongations of a fibred manifold. In this respect, a compilation of our 2-fibred manifold approach with the methods of natural operators theory is used.
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