Curvature structure of self-dual 4-manifolds

TL;DR

This paper explores the algebraic and geometric properties of self-dual 4-manifolds with signature (2,2), introducing new characterizations of their curvature tensors and advancing understanding of Osserman manifolds.

Contribution

It establishes a modified Cliff(1,1) structure compatible with Osserman models and provides new characterizations of the Weyl curvature tensor in self-dual 4-manifolds.

Findings

Existence of a modified Cliff(1,1) structure compatible with Osserman 0-models

New characterization of the Weyl curvature tensor in (anti-)self-dual manifolds

Additional results on (Jordan) Osserman manifolds

Abstract

We show the existence of a modified Cliff(1,1) structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we prove some new results regarding (Jordan) Osserman manifolds.