**MSC2010** 53C12, 57R30

### Riemannian foliations with Ehresmann connection

#### N. I. Zhukova^{1}

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

^{1}**Nina 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: http://orcid.org/0000-0002-4553-559X, nzhukova@hse.ru

**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