Reeb vector field

Template:More citations needed 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 $R\in \mathrm{k}\mathrm{e}\mathrm{r}d\alpha ,\alpha (R)=1$,
• 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

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