1 LINEAR CONNECTIONS AND CURVATURE TENSORS IN THE GEOMETRY OF PARALLELIZABLE MANIFOLDS

Dedicated to the memory of Prof. Dr. A. TAMIM Abstract. In this paper we discuss linear connections and curvature tensors in the context of geometry of parallelizable manifolds (or 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 semisymmetric. 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 semisymmetric 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 pointed out. 1