Collapse of warped foliations

Abstract

The aim of this note is to generalize the concept of warped product to a foliated manifold ( M , F , g ) as follows: If f : M → ( 0 , ∞ ) is a smooth function constant along the leaves of the foliation F then new metric structure g f on the manifold M is constructed as follows: g f ( v , w ) = f 2 g ( v , w ) if v , w are tangent to F and g f ( v , w ) = g ( v , w ) if v or w is perpendicular to F . A foliated manifold ( M , F , g f ) is called warped foliation while f is called warping function . Next, if ( f n : M → ( 0 , ∞ ) ) n ∈ N is a sequence of warping functions on M , the question of the existence of the limit in Gromov–Hausdorff of a sequence ( ( M , F , g f n ) ) n ∈ N warped foliation is asked. A number of examples is considered such foliations with dense leaf or foliations consisting of finite number of Reeb components. Next, sufficient and necessary condition of converging in Gromov–Hausdorff sense of a Riemannian foliation with all leaves compact to the space of leaves with a metric defined by Hausdorff distance of leaves is developed. Finally some results on Hausdorff foliations with all leaves compact are shown.