Reeb vector field explained

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

\alpha , the Reeb vector field satisfies R \in ker d\alpha, \alpha (R) = 1 ,^{[1] [2]}

• in particular, in the context of Sasakian manifold.

Definition

Let

\xi be a contact vector field on a manifold M of dimension 2 n +1 . Let \xi = Ker \alpha for a 1-form \alpha on M such that \alpha \wedge (d \alpha) ^{n} ≠ 0 . Given a contact form \alpha , there exists a unique field (the Reeb vector field) X _{\alpha} on M such that:^[3] i(X _{\alpha)d} \alpha = 0 i(X _{\alpha)} \alpha = 1 .

See also

• Weinstein conjecture

References

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

Notes and References

1. http://people.math.gatech.edu/%7Eetnyre/preprints/papers/phys.pdf
2. http://www2.im.uj.edu.pl/katedry/K.G/AutumnSchool/Monday.pdf
3. C. Vizman, "Some Remarks on the Quantomorphism Group" (1997)