In 1960, R. Hermann showed the following:
Theorem Let $ M $ to be a variety with a foliation $ F $ and a package-like metric, if all the sheets are compact and the holonomy group of each sheet is trivial, then $ M / F $ is a smooth variety.
(This is the partial result of the main theorem on Hermann, R., On the differential geometry of foliations, Ann. Math. (2) 72, 445-457 (1960). ZBL0196.54204.)
Q If we abandon the condition of a package and admit the trivial holistic group, can we achieve the same result? C & # 39; is to say:
Let $ M $ to be a variety with a foliation $ F $if all the leaves are compact and diffeomorphistic from each other and the holonomy group of each leaf is trivial, is it true that $ M / F $ is a smooth variety?
PS: I'm not sure if such a problem has been proven or refuted? Any reference is welcome.