Almost Kähler Eight-Dimensional Walker Manifold

Abstract

A Walker n-manifold is a pseudo-Riemannian manifold which admits a field of parallel null r-planes, with r ≤ n/2 . The canonical forms of the metrics were investigated by A. G. Walker [13]. Of special interest are the even-dimensional Walker manifolds (n = 2m) with fields of parallel null planes of half dimension (r = m). In this paper, we investigate geometric properties of some curvature tensors of an eightdimensional Walker manifold. Theorems for the metric to be Einstein, locally conformally at and for the Walker eight-manifold to admit a Kahler structure are given.