## A few remarks on the geometry of the space of leaf closures of a Riemannian foliation

### Volume 76 / 2007

#### Abstract

The space of the closures of leaves of a Riemannian foliation is a nice topological space, a stratified singular space which can be topologically embedded in $\mathbb{R}^k$ for $k$ sufficiently large. In the case of Orbit Like Foliations (OLF) the smooth structure induced by the embedding and the smooth structure defined by basic functions is the same. We study geometric structures adapted to the foliation and present conditions which assure that the given structure descends to the leaf closure space. In Section 5 we introduce the notion of an Ehresmann connection on a stratified foliated space and study the properties of the strata which depend on the existence of such a connection. We also give conditions which ensure that a connection understood as a differential operator defines an Ehresmann connection as above. In the last section we present some curvature estimates for metric structures on the leaf closure space.