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Prolongations and fields of paths for higher-order O.D.E. represented by connections on a fibered manifold.

Alexandr Vondra — 1996

Extracta Mathematicae

Higher-order differential equations represented by connections on prolongations of a fibered manifold.

Alexandr Vondra — 2000

Extracta Mathematicae

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.

Symmetries of connections on fibered manifolds

Alexandr Vondra — 1994

Archivum Mathematicum

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.

Sprays and homogeneous connections on $𝐑 \times 𝑇𝑀$

Alexandr Vondra — 1992

Archivum Mathematicum

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 $R \times T M$ - are studied. A natural correspondence between sprays and a special class of homogeneous connections is presented.

Natural dynamical connections

Alexandr Vondra — 1991

Czechoslovak Mathematical Journal

Geometry of second-order connections and ordinary differential equations

Alexandr Vondra — 1995

Mathematica Bohemica

The geometry of second-order systems of ordinary differential equations represented by $2$ -connections on the trivial bundle $error ℝ \times M \to ℝ$ 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...

Invariant subspaces in higher order jet prolongations of a fibred manifold

Miroslav Doupovec; Alexandr Vondra — 2000

Czechoslovak Mathematical Journal

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.

Some natural operations between connections on fibred manifolds

Doupovec, Miroslav; Vondra, Alexandr — 1996

Proceedings of the Winter School "Geometry and Physics"

Given a fibered manifold $Y \to X$ , a 2-connection on $Y$ means a section $J^{1} Y \to J^{2} Y$ . The authors determine all first order natural operators transforming a 2-connection on $Y$ and a classical linear connection on $X$ into a connection on $J^{1} Y \to Y$ . (The proof implies that there is no first order natural operator transforming 2-connections on $Y$ into connections on $J^{1} Y \to Y$ .) Using this result, the authors deduce several properties of characterizable connections on $J^{1} Y \to X$ .

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