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Article

Структура римановых слоений со связностью Эресмана

It is shown that the structural theory of Molino for Riemannian foliations on compact
manifolds and complete Riemannian manifolds is generalized to Riemannian foliations with
Ehresmann connection. There are no restrictions on the codimension of the foliation
and 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.
It is proved the equivalence of a number of properties of Riemannian foliations
$(M, F)$ with Ehresmann connection. 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 conformal foliations the similar statements are not true in general.