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
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.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
