Some properties concerning curvature tensors of eight-dimensional Walker manifolds (Q2912571)

A semi-Riemannian manifold \((M^n,g)\) is a Walker manifold if it has a field of parallel null \(r\)-planes with \(r\leq\frac{n}{2}\), i.e., there is a subbundle of \(TM\) of rank \(r\) to which the restriction of \(g\) vanishes identically. The paper under review is concerned with a special subclass of Walker manifolds of dimension \(n=8\) and maximal rank (\(r=4\)). Various geometric quantities are computed explicitly in terms of metric coefficients for this class, including the Christoffel symbols, curvature operator and Weyl tensor. Using these formulas, the authors write down the corresponding equations that imply that \(g\) is Einstein, Osserman, or locally conformally flat.