Foliations by minimal surfaces and contact structures on certain closed 3-manifolds
Let (M,g) be a closed, connected, oriented C∞ Riemannian 3-manifold with tangentially oriented flow F. Suppose that F admits a basic transverse volume form μ and mean curvature one-form κ which is horizontally closed. Let {X,Y} be any pair of basic vector fields, so μ(X,Y)=1. Suppose further that th...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2003-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S016117120320716X |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | Let (M,g) be a closed, connected, oriented C∞
Riemannian 3-manifold with tangentially oriented flow F. Suppose that F admits a basic transverse
volume form μ and mean curvature one-form κ which is horizontally closed. Let {X,Y} be any pair of basic vector fields, so μ(X,Y)=1. Suppose further that the globally defined vector 𝒱[X,Y] tangent to the flow satisfies
[Z.𝒱[X,Y]]=fZ𝒱[X,Y] for any basic vector
field Z and for some function fZ depending on Z. Then, 𝒱[X,Y] is either always zero and H, the distribution orthogonal to the flow in T(M), is integrable with minimal leaves, or 𝒱[X,Y] never vanishes and H is a contact structure. If additionally, M has a finite-fundamental group, then 𝒱[X,Y] never vanishes
on M, by the above together with a theorem of Sullivan (1979).
In this case H is always a contact structure. We
conclude with some simple examples. |
|---|---|
| ISSN: | 0161-1712 1687-0425 |