Twistor Bundle of a Neutral Kähler Surface

Włodzimierz Jelonek

doi:10.3390/sym14010043

Abstract

In this paper, we characterize neutral Kähler surfaces in terms of their positive twistor bundle. We prove that an O+,+(2,2)-oriented four-dimensional neutral semi-Riemannian manifold (M,g) admits a complex structure J with ΩJ∈⋀−M, such that (M,g,J) is a neutral-Kähler manifold if and only if the twistor bundle (Z1(M),gc) admits a vertical Killing vector field.

Highlights

A twistor bundle Z ± ( M) = {ω ∈ ± ( M) : g(ω, ω ) = 2} of a Riemannian fourmanifold ( M, g) was first defined and studied by Atiyah, Hitchin and Singer in [1]
We can define Z ± ( M ) as a one-parameter family gc of Riemannian metrics naturally induced by g and the Levi-Civita connection of g
We proved in [6] that a Riemannian four-manifold ( M, g) is Kähler if and only if its positive twistor bundle admits a non-zero vertical Killing vector field

Summary

Introduction

Of vertical Killing vector fields g ⊂ iso( P− ( M ), gc ), and it is a S1 -principal bundle over the twistor bundle Z ( M ), such that the natural projection is a Riemannian submersion. We proved in [6] that a Riemannian four-manifold ( M, g) is Kähler if and only if its positive twistor bundle admits a non-zero vertical Killing vector field.

Results

Conclusion