Curvature properties of four-dimensional Walker metrics

Abstract

A Walker n-manifold is a semi-Riemannian manifold, which admits a field of parallel null r-planes, with . In the present paper we study curvature properties of a Walker 4-manifold (M, g) which admits a field of parallel null 2-planes. The metric g is necessarily of neutral signature (+ + − −). Such a Walker 4-manifold is the lowest dimensional example not of Lorentz type. There are three functions of coordinates which define a Walker metric. Some recent work shows that a Walker 4-manifold of restricted type whose metric is characterized by two functions exhibits a large variety of symplectic structures, Hermitian structures, Kähler structures, etc. For such a restricted Walker 4-manifold, we shall study mainly curvature properties, e.g., conditions for a Walker metric to be Einstein, Osserman, or locally conformally flat, etc. One of our main results is the exact solutions to the Einstein equations for a restricted Walker 4-manifold.

Four-dimensional Osserman metrics with nondiagonalizable Jacobi operators

“…The fact that all known examples of nonsymmetric Jordan-Osserman metrics had nilpotent Jacobi operators of degree two or three suggested that no other examples could exist (cf. [7], [12], [16]). However, very recently the authors have shown the existence of a family of Type II Jordan-Osserman metrics with non-nilpotent Jacobi operators [10].…”

Some properties of four-dimensional Walker manifolds

“…In [3], Einsteinian, Osserman or locally conformally flat Walker manifolds were investigated in the restricted form of metrics when c(u 1 , u 2 , u 3 , u 4 ) = 0. In this paper, following [3], we consider the specific Walker metrics on a 4-dimensional manifold with…”

“…Among these, the significant Walker manifolds are examples of the non-symmetric and non-homogeneous Osserman manifolds [2,3]. Recently, it was shown [4,6,7] that the Walker 4-manifolds of neutral signature admit a pair comprising an almost complex structure and an opposite almost complex structure, and that Petean's nonflat indefinite Kahler-Einstein metric on a torus was obtained as an example of a Walker 4-manifold.…”

Stanilov-Tsankov-Videv Theory

“…The geometry of Walker manifolds with g 34 = 0 has been studied in [11]. We impose a different condition by setting g 33 = g 44 = 0 so the non-zero components of the metric are given by:…”