ISSN 2079-6900 (Print) 
ISSN 2587-7496 (Online)

Middle Volga Mathematical Society Journal

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MSC2010 53C12, 57R30

Riemannian foliations with Ehresmann connection

N. I. Zhukova1

AnnotationIt 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.
KeywordsRiemannian foliation, Ehresmann connection, local stability of a leaf, minimal set

1Nina I. Zhukova, Professor, Department of Fundamental Mathematics, Higher School of Economics (25/12, Bolshaya Pecherskaya St., Nizhni Novgorod 603155, Russia), Dr.Sci. (Physics and Mathematics), ORCID:,

Citation: N. I. Zhukova, "[Riemannian foliations with Ehresmann connection]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,20:4 (2018) 395–407 (In Russian)

DOI 10.15507/2079-6900.20.201804.395-407