On bicomplex-holomorphic manifolds (Q2845344)

A Walker four-manifold is a four-dimensional pseudo-Riemannian manifold of signature \((2,2)\) that admits a two-dimensional distribution that is light-like and parallel. A Walker four-manifold comes with a set of linear transformations on the tangent bundle that carries the structure of a bicomplex algebra. It is the purpose of the paper to study Walker four-manifolds that admit an additional metric with certain compatibility properties, a so-called double Norden metric, and to demonstrate that, under the given assumptions, the curvature tensor of the double Norden metric has to vanish.