On affine symmetries in 4-dimensional spaces of neutral signature

Abstract

The goal of this paper is to examine affine vector fields on $4-$dimensional spaces endowed with a metric of signature $(+, +, -, -)$. The existence of affine symmetries is discussed on these spaces and its connection between the holonomy structure is presented. The holonomy types of such spaces admitting proper affine vector fields are obtained and some examples are given. Additionally, several remarks are made regarding the close relationship of symmetries to Ricci solitons.