Some Geometric Properties of a Non-Strict Eight Dimensional Walker Manifold

An 8 dimensional Walker manifold (M; g) is a strict walker manifold if we can choose a coordinate system fx1; x2; x3; x4; x5; x6; x7; x8g on (M,g) such that any function f on the manfold (M,g), f(x1; x2; x3; x4; x5; x6; x7; x8) = f(x5; x6; x7; x8): In this work, we dene a Non-strict eight dimensional walker manifold as the one that we can choose the coordinate system such that for any f in (M; g); f(x1; x2; x3; x4; x5; x6; x7; x8) = f(x1; x2; x3; x4): We derive cononical form of the Levi-Civita connection, curvature operator, (0; 4)-curvature tansor, the Ricci tensor, Weyl tensorand study some of the properties associated with the class of Non-strict 8 dimensionalWalker manifold. We investigate the Einstein property and establish a theorem for the metric to be locally conformally at.

Para-Kähler–Einstein structures on Walker 4-manifolds

A 4-dimensional Walker manifold [Formula: see text] is a semi-Riemannian manifold [Formula: see text] of signature (++––) (or neutral), which admits a field of null 2-plane. The goal of this paper is to study certain almost paracomplex structures [Formula: see text] on 4-dimensional Walker manifolds. We discuss when these structures are integrable and when the para-Kähler forms are symplectic. We show that such a Walker 4-manifold can carry a class of indefinite para-Kähler–Einstein 4-manifolds, examples of indefinite para-Kähler 4-manifolds, and also almost indefinite para-Hermitian–Einstein 4-manifold. Finally, we give a counterexample for the almost para-Hemitian version of Goldberg conjecture.

Some General New Einstein Walker Manifolds

Lie symmetry group method is applied to find the Lie point symmetry group of a system of partial differential equations that determines general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.

A NOTE ON THE GOLDBERG CONJECTURE OF WALKER MANIFOLDS

This paper is concerned with Goldberg conjecture. Using the ϕφ-operator we prove the following result. Let (M, φ, w g) be an almost Kähler–Walker–Einstein compact manifold with the proper almost complex structure φ. The proper almost complex structure φ on Walker manifold (M, w g) is integrable if ϕφgN+ = 0, where gN+ is the induced Norden–Walker metric on M. This resolves a conjecture of Goldberg under the additional restriction on Norden–Walker metric (gN+ ∈ Ker ϕφ).