Complex surfaces and null conformal Killing vector fields

TL;DR

This paper explores the relationship between null conformal Killing vector fields and complex structures on pseudo-Riemannian manifolds, classifying certain four-manifolds and providing examples of neutral metrics with specific null Killing fields.

Contribution

It establishes the topological types of pseudo-Hermitian surfaces with null vector fields and classifies four-manifolds admitting pairs of null conformal Killing vector fields, introducing new structural insights.

Findings

Classification of pseudo-Hermitian surfaces with null vector fields

Identification of para-hyperhermitian structures from null Killing fields

Examples of neutral metrics with null Killing vector fields

Abstract

We study the relation between the existence of null conformal Killing vector fields and existence of compatible complex and para-hypercomplex structures on a pseudo-Riemannian manifold with metric of signature (2,2). We establish first the topological types of pseudo-Hermitian surfaces admitting a nowhere vanishing null vector field. Then we show that a pair of orthogonal, pointwise linearly independent, null, conformal Killing vector fields defines a para-hyperhermitian structure and use this fact for a classification of the smooth compact four-manifolds admitting such a pair of vector fields. We also provide examples of neutral metrics with two orthogonal, pointwise linearly independent, null Killing vector fields on most of these manifolds.