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...
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| Format: | Article |
| Language: | English |
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Wiley
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S016117120320716X |
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| author | Richard H. Escobales |
| author_facet | Richard H. Escobales |
| author_sort | Richard H. Escobales |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-0a2f6629869342b4b93d4f349f7e004e |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2003-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-0a2f6629869342b4b93d4f349f7e004e2025-02-03T06:00:44ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252003-01-012003211323133010.1155/S016117120320716XFoliations by minimal surfaces and contact structures on certain closed 3-manifoldsRichard H. Escobales0Department of Mathematics and Statistics, Canisius College, Buffalo 14208, NY, USALet (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.http://dx.doi.org/10.1155/S016117120320716X |
| spellingShingle | Richard H. Escobales Foliations by minimal surfaces and contact structures on certain closed 3-manifolds International Journal of Mathematics and Mathematical Sciences |
| title | Foliations by minimal surfaces and contact structures on certain closed 3-manifolds |
| title_full | Foliations by minimal surfaces and contact structures on certain closed 3-manifolds |
| title_fullStr | Foliations by minimal surfaces and contact structures on certain closed 3-manifolds |
| title_full_unstemmed | Foliations by minimal surfaces and contact structures on certain closed 3-manifolds |
| title_short | Foliations by minimal surfaces and contact structures on certain closed 3-manifolds |
| title_sort | foliations by minimal surfaces and contact structures on certain closed 3 manifolds |
| url | http://dx.doi.org/10.1155/S016117120320716X |
| work_keys_str_mv | AT richardhescobales foliationsbyminimalsurfacesandcontactstructuresoncertainclosed3manifolds |