It is well known that the concept of holonomic -jet can be geometrically characterized in terms of the contact of individual curves. However, this is not true for the semiholonomic -jets, [5], [8]. In the present paper, we discuss systematically the semiholonomic case.
We recall several different definitions of semiholonomic jet prolongations of a fibered manifold and use them to derive some interesting properties of prolongation of a first order connection to a third order semiholonomic connection.
The descriptions of Weil bundles, lifts of functions and vector fields are given. Actions of the automorphisms group of the Whitney sum of algebras of dual numbers on a Weil bundle of the first order are defined.
Orbits of complete families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows a description of the structure of the reduced phase space of a Hamiltonian system in terms of the reduced Poisson algebra. Moreover, one can give a global description of smooth geometric structures on a family of manifolds, which form a singular foliation of a subcartesian space, in terms of objects defined on the corresponding family of vector fields. Stratified...
Let be a principal bundle of frames with the structure group . It is shown that the variational problem, defined by -invariant Lagrangian on , can be equivalently studied on the associated space of connections with some compatibility condition, which gives us order reduction of the corresponding Euler-Lagrange equations.
Let F be a singular Riemannian foliation on a compact connected Riemannian manifold M. We demonstrate that global foliated vector fields generate a distribution tangent to the strata defined by the closures of leaves of F and which, in each stratum, is transverse to these closures of leaves.
The aim of the present work is to present a geometric formulation of higher order variational problems on arbitrary fibred manifolds. The problems of Engineering and Mathematical Physics whose natural formulation requires the use of second order differential invariants are classic, but it has been the recent advances in the theory of integrable non-linear partial differential equations and the consideration in Geometry of invariants of increasingly higher orders that has highlighted the interest...