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Reeb vector field

From Wikipedia, the free encyclopedia

In mathematics, the Reeb vector field, named after the French mathematician Georges Reeb, is a notion that appears in various domains of contact geometry including:

  • in a contact manifold, given a contact 1-form , the Reeb vector field satisfies ,[1][2]
  • in particular, in the context of Sasakian manifold.

Definition

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Let be a contact vector field on a manifold of dimension . Let for a 1-form on such that . Given a contact form , there exists a unique field (the Reeb vector field) on such that:[3]

.

See also

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References

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  • Blair, David E. (2010). Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics. Vol. 203 (Second edition of 2002 original ed.). Boston, MA: Birkhäuser Boston, Ltd. doi:10.1007/978-0-8176-4959-3. ISBN 978-0-8176-4958-6. MR 2682326. Zbl 1246.53001.
  • McDuff, Dusa; Salamon, Dietmar (2017). Introduction to symplectic topology. Oxford Graduate Texts in Mathematics (Third edition of 1995 original ed.). Oxford: Oxford University Press. doi:10.1093/oso/9780198794899.001.0001. ISBN 978-0-19-879490-5. MR 3674984. Zbl 1380.53003.