Riemannian foliations with Ehresmann connection

N. I. Zhukova

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 4, pp. 395-407

Résumé

It is shown that the structural theory of Molino for Riemannian foliations on compact manifolds and complete Riemannian manifolds may be generalized to a Riemannian foliations with Ehresmann connection. Within this generalization there are no restrictions on the codimension of the foliation and on the dimension of the foliated manifold. For a Riemannian foliation $(M, F)$ with Ehresmann connection it is proved that the closure of any leaf forms a minimal set, the family of all such closures forms a singular Riemannian foliation $(M, \overline{F})$. It is shown that in $M$ there exists a connected open dense $\overline{F}$-saturated subset $M_0$ such that the induced foliation $(M_0, \overline{F}|_{M_0})$ is formed by fibers of a locally trivial bundle over some smooth Hausdorff manifold. The equivalence of some properties of Riemannian foliations $(M, F)$ with Ehresmann connection is proved. In particular, it is shown that the structural Lie algebra of $(M, F)$ is equal to zero if and only if the leaf space of $(M, F)$ is naturally endowed with a smooth orbifold structure. Constructed examples show that for foliations with transversally linear connection and for conformal foliations the similar statements are not true in general.

Détail

Keywords: Riemannian foliation, Ehresmann connection, local stability of a leaf, minimal set.