Related papers: Generalized forms, metrics and Ricci flat four-met…
The algebra and calculus of generalized differential forms are reviewed and employed to construct a class of generalized connections and to investigate their properties. The class includes generalized connections which are flat when…
A formulation of Einstein's gravitational field equations in four space-time dimensions is presented using generalized differential forms and Cartan's equations for metric geometries. Cartan's structure equations are extended by using…
Generalized differential forms are employed to construct generalized connections. Lorentzian four-metrics determined by certain of these connections satisfy Einstein's vacuum field equations when the connections are flat. Generalized…
Vector fields with components which are generalized zero-forms are constructed. Inner products with generalized forms, Lie derivatives and Lie brackets are computed. The results are shown to generalize previously reported results for…
A generalized symmetry of a system of differential equations is an infinitesimal transformation depending locally upon the fields and their derivatives which carries solutions to solutions. We classify all generalized symmetries of the…
We show that any solution of the 4D Einstein equations of general relativity in vacuum with a cosmological constant may be embedded in a solution of the 5D Ricci-flat equations with an effective 4D cosmological "constant" that is a specific…
We prove that the Einstein equations can be solved in a very general form for arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases following a geometric method of anholonomic frame deformations for constructing…
Lie derivatives of various geometrical and physical quantities define symmetries and conformal symmetries in general relativity. Thus we obtain motions, collineations, conformal motions and conformal collineations. These symmetries are used…
We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For…
We study an even dimensional manifold with a pseudo-Riemannian metric with arbitrary signature and arbitrary dimensions. We consider the Ricci flat equations and present a procedure to construct solutions to some higher (even) dimensional…
We show how the Einstein equations with cosmological constant (and/or various types of matter field sources) can be integrated in a very general form following the anholonomic deformation method for constructing exact solutions in four and…
Generalized symmetries of the Einstein equations are infinitesimal transformations of the spacetime metric that formally map solutions of the Einstein equations to other solutions. The infinitesimal generators of these symmetries are…
It is shown that for a wide class of analytic Lagrangians which depend only on the scalar curvature of a metric and a connection, the application of the so--called ``Palatini formalism'', i.e., treating the metric and the connection as…
This review examines the role of differential forms, Pfaffian systems, and hypersurfaces in general relativity. These mathematical constructions provide the essential tools for general relativity, in which the curvature of…
For a generalized soliton $(g,\xi,\eta,\beta,\gamma,\delta)$, we provide necessary and sufficient conditions for the dual $1$-form $\xi^{\flat}$ of the potential vector field $\xi$ to be a solution of the Schr\"{o}dinger-Ricci equation, a…
An examples of a Ricci-flat of four-dimensional spaces with a Walker metrics and their generalizations are constructed. The properties of corresponding geodesic equations are discussed.
The physical consistency of the match of piecewise-$C^0$ metrics is discussed. The mathematical theory of gravitational discontinuity hypersurfaces is generalized to cover the match of regularly discontinuous metrics. The mean-value…
We review the language of differential forms and their applications to Riemannian Geometry with an orientation to General Relativity. Working with the principal algebraic and differential operations on forms, we obtain the structure…
Elie Cartan's general equivalence problem is recast in the language of Lie algebroids. The resulting formalism, being coordinate and model-free, allows for a full geometric interpretation of Cartan's method of equivalence via reduction and…
A generalization of the Einstein equation is considered for complex line elements. Several second order semilinear partial differential equations are derived from it as semilinear field equations in uniform and isotropic spaces. The…