Jan Brajercik

This paper is devoted to the geometric theory of a Schwarzschild spacetime, a basic objective in applications of the classical general relativity theory. In a broader sense, a Schwarzschild spacetime is a smooth manifold, endowed with an action of the special orthogonal group SO(3) and a Schwarzschild metric, an SO(3)-invariant metric field, satisfying the Einstein equations. We prove the existence of and find all Schwarzschild metrics on two topologically non-equivalent manifolds, R×(R3∖{(0,0,0)}) and S1×(R3∖{(0,0,0)}). The method includes a classification of SO(3)-invariant, time-translation invariant and time-reflection invariant metrics on R×(R3∖{(0,0,0)}) and a winding mapping of the real line R onto the circle S1. The resulting family of Schwarzschild metrics is parametrized by an arbitrary function and two real parameters, the integration constants. For any Schwarzschild metric, one of the parameters determines a submanifold, where the metric is not defined, the Schwarzschild...

In this paper, we introduce the structure of a principal bundle on the r-jet prolongation J r FX of the frame bundle FX over a manifold X. Our construction reduces the well-known principal prolongation W r FX of FX with structure group G nr. For a structure group of J r FX we find a suitable subgroup of G nr. We also discuss the structure of the associated bundles. We show that the associated action of the structure group of J r FX corresponds with the standard actions of differential groups on tensor spaces.

The multiple-integral variational functionals for finite-dimensional immersed submanifolds are studied by means of the fundamental Lepage equivalent of a homogeneous Lagrangian, which can be regarded as a generalization of the well-known Hilbert form in the classical mechanics. The notion of a Lepage form is extended to manifolds of regular velocities and plays a basic role in formulation of the variational theory for submanifolds. The theory is illustrated on the minimal submanifolds problem, including analysis of conservation law equations.

Variational principles on frame bundles, invariant with respect to the structure group are investigated. Explicit expressions for the first-order invariant lagrangians, the Poincaré-Cartan, and the Euler-Lagrange forms are found, and the corresponding conservation laws are obtained as a consequence of the Noether’s theorem. We show that the (second-order) Euler-Lagrange equations for a frame field are equivalent with the system of (first-order) conservation laws for this field. Analogous results are obtained for second-order invariant Lagrangians on frame bundles.

The aim of this paper is to characterize all second order tensor-valued and scalar differential invariants of the bundle of linear frames F X over an n-dimensional manifold X. These differential invariants are ob-tained by factorization method and are described in terms of bases of invariants. Second order natural Lagrangians of frames have been charac-terized explicitly; if n = 1, 2, 3, 4, the number of functionally independent second order natural Lagrangians is N = 0, 6, 33, 104, respectively.

We present the theory of first order local variational principles in fibered manifolds, in which the fundamental global concept is a locally variational dynamical form. The resulting theory of extremals and symmetries is also discussed. Using any Lepage form, our approach differs from Prieto, who used the Poincaré-Cartan form for definiton of a local variational principle.

ABSTRACT Let μ:FX→X be a principal bundle of frames with the structure group Gln(R) and let λ be a Gln(R)-invariant Lagrangian on J1FX. We give an explicit expressions of reduced equations for the associated sections of the corresponding bundle of linear connections which replace the Euler–Lagrange equations for the variational problem defined by λ.

Variational principles on frame bundles, given by the first and the second order Lagrangians invariant with respect to the struc-ture group, are considered. Noether's currents, associated with the corre-sponding Lepage equivalents, are obtained. It is shown that for the first and the second order invariant variational problems, the system of the Euler-Lagrange equations for a frame field are equivalent with the lower order system of equations.