Four-dimensional Osserman metrics of neutral signature

TL;DR

This paper characterizes and classifies four-dimensional neutral signature Osserman metrics, establishing their equivalence in algebraic models and identifying geometric conditions for null Jordan Osserman manifolds.

Contribution

It proves the equivalence of null, spacelike, and timelike Osserman conditions in signature (2,2) and classifies null Jordan Osserman models in this setting.

Findings

Null, spacelike, and timelike Osserman are equivalent in signature (2,2)

Classification of null Jordan Osserman models in signature (2,2)

Null Jordan Osserman manifolds are either constant curvature or complex space forms

Abstract

In the algebraic context, we show that null Osserman, spacelike Osserman, and timelike Osserman are equivalent conditions for a model of signature (2,2). We also classify the null Jordan Osserman models of signature (2,2). In the geometric context, we show that a pseudo-Riemannian manifold of signature (2,2) is null Jordan Osserman if and only if either it has constant sectional curvature or it is locally a complex space form.