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 $\alpha$, the Reeb vector field satisfies $$\begin{gathered} R\in \mathrm{k}\mathrm{e}\mathrm{r} \\ d\alpha , \\ \alpha (R)=1 \end{gathered}$$,
• in particular, in the context of Sasakian manifold.

Definition

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

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

.

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.