Linear connexions with zero torsion and recurrent curvature

Abstract

Introduction. Let M be a connected n-dimensional C4-manifold. If S is any tensor (field) on M, then (i) S(u) denotes the value of S at the point uGM; (ii) S =0 means that S is everywhere zero, i.e. S(u) =0 for every u CM; (iii) S $0 means that S is not everywhere zero, i.e. S(u) 0 for some u CM; (iv) S$O means that S is nowhere zero, i.e. S(u) =0 for no uCM. Throughout this paper, a tensor of type (1, 0) will be called a vector, and a tensor of type (0, 1) a covector. Summation over a repeated index, Latin or Greek, is always implied. Let y be a linear connexion on M. If rFJh (1_ a, h, i, j, * , ?n) are the components of y in the local coordinate system (U, uh) in M, then the components in (U, uh) of the torsion tensor T and the curvature tensor R on M are respectively