Examples of signature (2,2) manifolds with commuting curvature operators

TL;DR

This paper constructs specific Walker manifolds of signature (2,2) demonstrating different commutativity properties among curvature-related operators, linking these properties to underlying affine structures.

Contribution

It provides explicit examples of signature (2,2) Walker manifolds with various curvature operator commutativity properties, connecting them to affine structures.

Findings

Constructed Walker manifolds with specific curvature operator properties

Linked curvature properties to Ricci tensor of underlying affine structure

Expanded understanding of curvature operator interactions in signature (2,2) manifolds

Abstract

We exhibit Walker manifolds of signature (2,2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure A, these properties are related to the Ricci tensor of A.