Transversal Twistor Spaces of Foliations

TL;DR

This paper explores the geometry of transversal twistor spaces of foliations, linking the existence of certain complex structures to the properties of Bott connections and the transversally projective nature of the foliation.

Contribution

It establishes conditions under which transversal almost complex structures are projectable and relates these to the flatness of the transversal projective structure.

Findings

Existence of projectable structures I and J depends on transversally projective foliation.

Structure J is never integrable, while I's integrability depends on flatness of the transversal structure.

Provides cohomological conditions for the existence of projectable structures.

Abstract

The transversal twistor space of a foliation F of an even codimension is the bundle ZF of the complex structures of the fibers of the transversal bundle of F. On ZF, there exists a foliation F' by covering spaces of the leaves of F, and any Bott connection of F produces an ordered pair (I,J) of transversal almost complex structures of F'. The existence of a Bott connection which yields a structure I that is projectable to the space of leaves is equivalent to the fact that F is a transversally projective foliation. A Bott connection which yields a projectable structure J exists iff F is a transversally projective foliation which satisfies a supplementary cohomological condition, and, in this case, I is projectable as well. J is never integrable. The essential integrability condition of I is the flatness of the transversal projective structure of F.