Extended Absolute Parallelism Geometry

TL;DR

This paper introduces an extended version of Absolute Parallelism geometry on tangent bundles, incorporating directional dependence and new geometric objects, with potential applications in generalized field theories.

Contribution

It develops the framework of Extended Absolute Parallelism geometry, including new geometric objects, curvature formulas, and conditions, extending classical AP-geometry to a more general setting.

Findings

Derived explicit curvature and W-tensors in EAP-geometry.

Identified conditions under which EAP reduces to classical AP-geometry.

Discussed physical implications and potential for generalized field theories.

Abstract

In this paper, we study Absolute Parallelism (AP-) geometry on the tangent bundle $TM$ of a manifold $M$. Accordingly, all geometric objects defined in this geometry are not only functions of the positional argument $x$, but also depend on the directional argument $y$. Moreover, many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this different framework. We refer to such a geometry as an Extended Absolute Parallelism (EAP-) geometry. The building blocks of the EAP-geometry are a nonlinear connection assumed given a priori and $2n$ linearly independent vector fields (of special form) defined globally on $TM$ defining the parallelization. Four different $d$-connections are used to explore the properties of this geometry. Simple and compact formulae for the curvature tensors and the W-tensors of the four defined $d$-connections are obtained, expressed in terms of the torsion and the contortion tensors of the EAP-space. Further conditions are imposed on the canonical $d$-connection assuming that it is of Cartan type (resp. Berwald type). Important consequences of these assumptions are investigated. Finally, a special form of the canonical $d$-connection is studied under which the classical AP-geometry is recovered naturally from the EAP-geometry. Physical aspects of some of the geometric objects investigated are pointed out and possible physical implications of the EAP-space are discussed, including an outline of a generalized field theory on the tangent bundle $TM$ of $M$