Reeb vector field

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 ${\displaystyle \alpha }$, the Reeb vector field satisfies ${\displaystyle R\in \mathrm {ker} \ d\alpha ,\ \alpha (R)=1}$,^[1][2]
• in particular, in the context of Sasakian manifold.

Definition

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

• ${\displaystyle i(X_{\alpha })d\alpha =0}$
• ${\displaystyle i(X_{\alpha })\alpha =1}$

.

See also

• Weinstein conjecture

References

• 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.