Miroslav Doupovec

Holonomic extension of connections and symmetrization of jets

Taking into account motivations from the geometrization of mathematical physics, we discuss the symmetrization of higher order jets and extension of connections. First we introduce an extension of a general connection Γ on a fibered manifold Y → M into an r -th order holonomic connection and we classify all second-order holonomic extensions of Γ. Then we study the general problem of symmetrization of jets. We introduce symmetrizations of higher-order semiholonomic and nonholonomic prolongations and we describe all symmetrizations of an r -th order semiholonomic prolongation j ¯ r .

Natural relations between connections in 2-fibred manifolds

In the paper, a new approach to the study of interrelations between connections within the framework of a general 2-fibred manifold is introduced. The role of a naturality of the constructions is discussed and the formalism is applied in two particular situations. First, natural relations between connections in J 1 Y→ Y → X are studied and their role for the integrability of partial differential equations represented by connections in question is described. Secondly, the 2-fibred manifold VY → Y → X is investigated. In this case, the importance for a description of the symmetries of equations under consideration is pointed out.

Invariant subspaces in higher order jet prolongations of a fibred manifold

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.