Leaves of Foliations with a Transverse Geometric Structure of Finite Type

ROBERT A. WOLAK In Chis short note we find some conditions which ensure that a G foliation of finite type with all leaves compact is a Riemannian foliation or equivalently the space of leaves of such a foliation is a Satake manifold . A particular attention is paid to transversely affine foliations . We present several conditions such ensure completeness of these foliations . In this short note we shall study leaves of foliations with some transverse geometric structure . In the first section we deal with G-foliations of finite type with compact leaves . The main point is to determine whether the space of leaves of such a foliation is a Satake manifold . In the second section we turn our attention to transversely affine foliations . The question is of importance as, if the space of leaves is a Satake manifold, the foliation itself is Riemannian . It is not a trivial problem. Even for transversely affine foliations with all leaves compact we do not know whether such a foliation is Riemannian . Of course, some additional conditions like completeness or distality are sufficient, but we have been unable to show that a transversely affine foliation with all leaves compact is complete . The author would like to thank very much indeed Marcel Nicolau for proposing the subject of this note and for many discussions which helped understand better the geometry of foliations in general and the foliations with compact leaves in particular . This note was partially written while the author was a guest of the Centre de Recerca Matematica . He is also very grateful to Professors Manuel Castellet and Joan Girbau for the invitation and the creation of most agreable atmosphere for work . Finally, the author would like to express his deep gratitude to Professor Gilbert Hector whose remarks and advice helped ameliorate a great deal the first version of the paper . 1 . Let F be a G-foliation . The bundle L(M; .F) of transverse linear frames admits a foliated reduction B(M, G; .'F) to the group G. The foliation F is called of finite type k if the Lie group G is of type k. This means that the foliation .Fk on the total space of the (k 1) -th prolongation Bk-1(M, G; .F) of the foliated G-structure B(M, G; .'F) is a T.P . foliation, cf. [16] and [20] . The leaves of .Fk are converings of leaves of .F . Therefore the following is true .