The geometry of the space of leaf closures of a transversely almost Kähler foliation

We study the geometry of the leaf closure space of regular and singular Riemannian foliations. We give conditions which assure that this leaf space is a singular symplectic or Kähler space. In recent years physicists and mathematicians working on mathematical models of physical phenomena have realised that modelling based on geometric structures on now classical smooth manifolds is insufficient. In some cases more complicated topological spaces appear naturally and one would like to develop geometry of such spaces. One of the well-known examples is the orbit space of a smooth action of a compact Lie group. Such a space is a stratified pseudomanifold of Goresky-MacPherson, cf. [7, 9, 4]. This fact has been used to describe the topology and structure of the reduced space of the momentum map in the singular case, cf. [15]. The study of the Riemannian geometry of the orbit space of a smooth action of a compact Lie group has been initiated in [1]. One should also mention K. Richardson’s paper, cf. [13], in which the author demonstrates that any space of orbits of such an action is homeomorphic to the space of the closures of leaves of a regular Riemannian foliation, and, obviously, cf. [11], vice versa. In this paper we innitiate the study of the geometry of the space of leaf closures of a Riemannian foliation, both regular or singular. The first part is concerned with the space of leaf closures of an interesting class of regular Riemannian foliations transversely almost hermitian, which includes transversely almost Kähler foliations. In this case the foliated manifold is presymplectic. The aim of this note is to describe the geometry of the leaf space M/F ; in other word to describe the way in which the Riemannian, almost complex and symplectic structures descend onto this leaf space.