Paolo Piccinni, Izu Vaisman
Foliations with Transversal Quaternionic Structures
Preprint series: ESI preprints

MSC:

53C12 Foliations (differential geometric aspects), See Also {57R30, 57R32}

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

Abstract: We consider manifolds equipped with a foliation
$\cal F$ of codimension $4q$,
and an almost quaternionic structure $Q$ on the
transversal bundle of ${\cal F}$. After discussing conditions
of projectability and integrability of $Q$, we study the
transversal twistor space
$Z{\cal F}$ which, by definition, consists
of the $Q$-compatible almost complex
structures. We show that
$Z{\cal F}$ can be endowed with a lifted foliation
${\widehat {\cal F}}$ and two natural
almost complex structures $J_1$, $J_2$ on the transversal bundle of
$\widehat{\cal F}$.
We establish the conditions which ensure the
projectability of $J_1$ and $J_2$, and the integrability
of $J_{1}$ ($J_{2}$ is never integrable).


Keywords: Foliation, Quaternionic Structure, Twistor Space