• Source: 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



α


{\displaystyle \alpha }

, the Reeb vector field satisfies



R
∈

k
e
r


d
α
,

α
(
R
)
=
1


{\displaystyle R\in \mathrm {ker} \ d\alpha ,\ \alpha (R)=1}

,
in particular, in the context of Sasakian manifold.




Definition


Let



ξ


{\displaystyle \xi }

be a contact vector field on a manifold



M


{\displaystyle M}

of dimension



2
n
+
1


{\displaystyle 2n+1}

. Let



ξ
=
K
e
r

α


{\displaystyle \xi =Ker\;\alpha }

for a 1-form



α


{\displaystyle \alpha }

on



M


{\displaystyle M}

such that



α
∧
(
d
α

)

n


≠
0


{\displaystyle \alpha \wedge (d\alpha )^{n}\neq 0}

. Given a contact form



α


{\displaystyle \alpha }

, there exists a unique field (the Reeb vector field)




X

α




{\displaystyle X_{\alpha }}

on



M


{\displaystyle M}

such that:




i
(

X

α


)
d
α
=
0


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





i
(

X

α


)
α
=
1


{\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.

Kata Kunci Pencarian:

• Reeb vector field
• Contact geometry
• Sasakian manifold
• Reeb
• Weinstein conjecture
• Georges Reeb
• Floer homology
• Lagrangian Grassmannian
• Trivial cylinder
• Clifford Taubes