Linear Connections and Curvature Tensors in the Geometry of Parallelizable Manifolds

TL;DR

This paper explores curvature tensors in Absolute Parallelism geometry, expressing them via torsion tensors, deriving identities, and examining properties of the Wanas tensor, especially under semi-symmetric connection conditions, with implications for physics.

Contribution

It introduces a unified expression for curvature tensors in terms of torsion and analyzes the Wanas tensor's properties in semi-symmetric connections.

Findings

Curvature tensors expressed via torsion tensors.

Derived identities from Bianchi identities.

Under semi-symmetric conditions, curvature tensors align with the Wanas tensor.

Abstract

In this paper we discuss curvature tensors in the context of Absolute Parallelism geometry. Different curvature tensors are expressed in a compact form in terms of the torsion tensor of the canonical connection. Using the Bianchi identities some other identities are derived from the expressions obtained. These identities, in turn, are used to reveal some of the properties satisfied by an intriguing fourth order tensor which we refer to as Wanas tensor. A further condition on the canonical connection is imposed, assuming it is semi-symmetric. The formulae thus obtained, together with other formulae (Ricci tensors and scalar curvatures of the different connections admitted by the space) are calculated under this additional assumption. Considering a specific form of the semi-symmetric connection causes all nonvanishing curvature tensors to coincide, up to a constant, with the Wanas tensor. Physical aspects of some of the geometric objects considered are mentioned.