Abstract
The transversal twistor space of a foliation \(F\) of an even codimension is the bundle \(Z(F)\) of the complex structures of the fibers of the transversalbundle of \(F\). On \(Z(F)\)there exists a foliation \(\hat F\)by covering spaces of the leaves of \(F\), and any Bottconnection of \(F\) produces an ordered pair\((\ell _1 ,\ell _2 )\)of transversal almost complex structures of \(\hat F\). The existence of a Bott connection which yields a structure\(\ell \) 1 that is projectable to the space of leaves isequivalent to the fact that \(F\) is a transversallyprojective foliation. A Bott connection which yields a projectablestructure \(\ell \) 2 exists iff \(F\) isa transversally projective foliation which satisfies a supplementarycohomological condition, and, in this case, \(\ell \) 1is projectable as well. \(\ell \) 2 is never integrable.The essential integrability condition of \(\ell \) 1 isthe flatness of the transversal projective structure of \(F\).
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Vaisman, I. Transversal Twistor Spaces of Foliations. Annals of Global Analysis and Geometry 19, 209–234 (2001). https://doi.org/10.1023/A:1010721712802
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DOI: https://doi.org/10.1023/A:1010721712802