Transverse Equivalence of Complete Conformal Foliations

Abstract

We study the problem of classification of complete non-Riemannian conformal foliations of codimension q > 2 with respect to transverse equivalence. It is proved that two such foliations are transversally equivalent if and only if their global holonomy groups are conjugate in the group of conformal transformations of the q-dimensional sphere Conf (S q ). Moreover, any countable essential subgroup of the group Conf (S q ) is realized as the global holonomy group of some non-Riemannian conformal foliation of codimension q. Bibliography: 16 titles.

Dynamics of conformal foliations

“…In comparison with the notion of transverse equivalence introduced by P. Molino [20, p. 63 Using this additional requirement, we proved that the strong transverse equivalence of foliations covered by fibrations can be realized by a foliation covered by a fibration. This fact was used in the proofs of the following theorems in [30]. As we have shown, in general, this is not true for transverse equivalent foliations in the sense of P. Molino.…”

“…The notion of strong transverse equivalence of foliations was introduced and investigated by us in [30] under the name "transverse equivalence".…”