Ivan Kolář

A geometrical version of the higher order Hamilton formalism in fibred manifolds

Abstract It has been clarified recently that an r-th order Lagrangian on a fibred manifold Y → X does not determine a unique Poincare-Cartan form provided dim X > 1 and r > 2, [1], [4], [6], [9], [10]. To make this fact more transparent, we introduced a new operation generalizing the formal exterior differentiation, [6]. In the present paper we deduce in such a way that a unique Poincare-Cartan form can be determined by means of a simple additional structure - a linear symmetric connection Г on the base manifold X (or, more generally, by a convenient splitting S). Then we present a suitable geometric definition of a regular r-th order Lagrangian on Y and we prove that any our Poincare-Cartan form can be used in a geometrical version of the higher order Hamilton formalism.

On the fiber product preserving bundle functors

Abstract We present a complete description of all fiber product preserving bundle functors on the category of fibered manifolds with m-dimensional bases and fiber preserving maps with local diffeomorphisms as base maps. This result is based on several general properties of such functors, which are deduced in the first two parts of the paper.

On the Helmholtz operator for Euler morphisms

The variational sequence describes the Helmholtz conditions for local variationality in terms of the Helmholtz map, which is defined on a factor space. We study a tensor modification called the Helmholtz operator. For the first and second order cases we prove that, up to a multiplicative constant, the Helmholtz operator is the unique natural operator of the type in question.

On the natural operators on vector fields

We determine all natural operators transforming every vector field on a manifold M into a vector field on FM, where F is any natural bundle corresponding to a product preserving functor.This research was finished during the authors stay at the University of Vienna. The author acknowledges its kind hospitality and is grateful to Peter Michor, Jan Slovák and Jiří Vanžura for several useful comments.All manifolds and maps are assumed to be infinitely differentiable.

Affine structure on Weil bundles

Abstract. For every r-th order Weil functor T(A), we introduce the underliyng k-th order Weil functors T(Ak), k=1,...,r-1. We deduce that T(A)M -> T(Ar-1)M is an affine bundle for every manifold M. Generalizing the classical concept of contakt element by C. Ehresmann, we define the bundle of contact elements of type A on M and we describe some affine properties of this bundle.

Torsion-free connections on higher order frame bundles

We deduce that torsion-free connections on the r-th order frame bundle P r M of a manifold M can be identified with certain reductions of P r+1 M. They are also interpreted as splittings of T*M into the bundle of all (1,r+l)-covelocities on M. Finally we determine all natural operators transforming torsion-free connections on P 1 M into torsion-free connections on P 2 M.

Some Gauge-natural operators on linear connections

We determine explicitly all geometrical operators transforming a linear connection on a vector bundle π:E→M and a classical linear connection on the base manifoldM into a classical linear connection on the total spaceE.

Contact elements on fibered manifolds

AbstractFor every product preserving bundle functor Tμ on fibered manifolds, we describe the underlying functor of any order (r, s, q), s ≥ r ≤ q. We define the bundle

On the torsion of linear higher order connections

K_{k,l}^{r,s,q} Y