Four-dimensional Osserman metrics of neutral signature

Abstract

In the algebraic context, we show 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 with this signature is null Jordan Osserman if and only if either it has constant sectional curvature or it is locally a complex space form.

Duality principle and special Osserman manifolds

“…It is worth to emphasize that the spacelike and timelike Osserman conditions at any point are equivalent [12]. However, an analogous result is no longer true for the spacelike and timelike Jordan-Osserman conditions, which are equivalent in dimension four [11].…”

Compact Osserman Manifolds with Neutral Metric

“…One says that M is Jordan-Osserman if the Jordan normal form of the Jacobi operators is constant on the pseudo-sphere bundles S ± (T M ). Although the spacelike and timelike Jordan-Osserman conditions are equivalent in signature (− − ++) they do not necessarily imply the null Jordan-Osserman condition (see [17] for details).…”

Curvature Structure of Self-Dual 4-Manifolds

“…Their exact definitions and other preliminaries can be found in Section 2. Section 3 is dedicated to the proof of the following proposition, which is our main algebraic result; this result plays a crucial role in the analysis of [7,12].…”