SOME GEOMETRIC PROPERTIES OF A FAMILY WALKER METRIC ON A EIGHT-DIMENSIONAL MANIFOLDS

SOME GEOMETRIC PROPERTIES OF A FAMILY WALKER...

Abstract

A pseudo-Riemannian manifold which admits a field of parallel
null r-planes, with r ≤ n 2 is a Walker n-manifold. A.G. Walker in [12] investi-
gated the canonical forms of the metrics and came out with some interesting re-
sults. 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
consider a perticular eight-dimensional Walker manifold, derive and investigate
some geometric properties of the curvature tensors of the Manifold. We give
some theorems for the metric to be Einstein and Locally Conformally Flat.