Computation of Partially Invariant Solutions for the Einstein Walker Manifolds' Identifying Equations

TL;DR

This paper applies the partially invariant solutions method to classify and compute new four-dimensional Einstein Walker manifolds, extending traditional similarity reduction techniques through subgroup classification of PDE symmetries.

Contribution

It introduces a novel application of PISs to Einstein Walker manifolds, demonstrating solutions that differ from those obtained by standard similarity reduction.

Findings

Derived new classes of Einstein Walker manifolds using PISs

Showed PISs can produce solutions distinct from similarity reduction

Extended the method of subgroup classification to this geometric context

Abstract

In this paper, partially invariant solutions (PISs) method is applied in order to obtain new four-dimensional Einstein Walker manifolds. This method is based on subgroup classification for the symmetry group of partial differential equations (PDEs) and can be regarded as the generalization of the similarity reduction method. For this purpose, those cases of PISs which have the defect structure delta=1 and are resulted from two-dimensional subalgebras are considered in the present paper. Also it is shown that the obtained PISs are distinct from the invariant solutions that obtained by similarity reduction method.