The transversal twistor space of a foliation ℱ of an even codimension is the bundle script Z sign(ℱ) of the complex structures of the fibers of the transversal bundle of ℱ. On script Z sign(ℱ) there exists a foliation ℱ̂ by covering spaces of the leaves of ℱ, and any Bott connection of ℱ̂ produces an ordered pair (ℓ1, ℓ2) of transversal almost complex structures of ℱ̂. The existence of a Bott connection which yields a structure ℓ1 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 ℓ2 exists iff ℱ is a transversally projective foliation which satisfies a supplementary cohomological condition, and, in this case, ℓ1 is projectable as well. ℓ2 is never integrable. The essential integrability condition of ℓ1 is the flatness of the transversal projective structure of ℱ.
Bibliographical noteFunding Information:
Part of this work was done during visits of the author to Istituto di Matematica, Università di Roma 1 and Dipartimento di Matematica, Università della Basilicata, Potenza, Italy (visit sponsored by the Consiglio Nazionale delle Ricerche, Italy), Centre de Mathématiques, École Polytechnique, Palaiseau, France, and the universities of Jassy and Bras¸ov (Romania). The author wishes to express here his gratitude to these institutions and to his hosts Paolo Piccinni, Sorin Dragomir, Yvette Kosmann-Schwarzbach, François Laudenbach, Paul Gauduchon, Radu Miron, Vasile Cruceanu, Vasile Oproiu, Mihai Anastasiei, and Gheorghe Munteanu.
- Foliated (projectable) objects
- Transversal twistor spaces
ASJC Scopus subject areas
- Political Science and International Relations
- Geometry and Topology