International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 21, Pages 1323-1330
Foliations by minimal surfaces and contact structures on certain closed -manifolds
Department of Mathematics and Statistics, Canisius College, Buffalo 14208, NY, USA
Received 14 July 2002; Revised 7 August 2002
Copyright © 2003 Richard H. Escobales. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be a closed, connected, oriented
Riemannian 3-manifold with tangentially oriented flow . Suppose that admits a basic transverse
volume form and mean curvature one-form which is horizontally closed. Let be any pair of basic vector fields, so . Suppose further that the globally defined vector tangent to the flow satisfies
for any basic vector
field and for some function depending on . Then, is either always zero and , the distribution orthogonal to the flow in , is integrable with minimal leaves, or never vanishes and is a contact structure. If additionally, has a finite-fundamental group, then never vanishes
on , by the above together with a theorem of Sullivan (1979).
In this case is always a contact structure. We
conclude with some simple examples.