Derivation of Field Equations in Space with the Geometric Structure Generated by Metric and Torsion

Abstract

This paper is devoted to the derivation of field equations in space with the geometric structure generated by metric and torsion tensors. We also study the geometry of the space generated jointly and agreed on by the metric tensor and the torsion tensor. We showed that in such space the structure of the curvature tensor has special features and for this tensor we obtained analog Ricci-Jacobi identity and evaluated the gap that occurs at the transition from the original to the image and vice versa, in the case of infinitely small contours. We have researched the geodesic lines equation. We introduce the tensor παβ which is similar to the second fundamental tensor of hypersurfaces Yn-1, but the structure of this tensor is substantially different from the case of Riemannian spaces with zero torsion. Then we obtained formulas which characterize the change of vectors in accompanying basis relative to this basis itself. Taking into considerations our results about the structure of such space we derived from the variation principle the general field equations (electromagnetic and gravitational).

The Dаrbоuх Theory and Geodesics in the Metric Space with Torsion

“…HIS work is dedicated to the theory of the n Y space, analytically, this space is an n-dimensional differentiable real manifold, at each point of which are given metric and torsion tensors [29][30][31][32][33]. From a geometrical perspective, such space can be defined as a real n-dimensional metric space equipped with a connection on the tangent bundle (that connection can be with torsion); this metric is generated by a given symmetrical covariant tensor, and the torsion of the connection of the space coincides with given torsion tensor [1][2][3][4][5][29][30][31][32][33].…”

Theory of Hypersurfaces yn–1 in yn Space and Geodesics on this Hypersurfaces yn–1

“…The hypersurface theory in the n Y space has attracted a lot of attention after its introduction [1,2], since it has many applications in theoretical and applied physics and electromagnetic gravity.…”

“…Intrinsic geometrical properties have been considered in [1][2][3][4][5][6][7][8][9] a space and their context and the relations with the ambient space n Y within [9].…”