Reeb vector field

Last updated April 16, 2024

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:

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 $\alpha \wedge (d\alpha )^{n}\neq 0$ . Given a contact form $\alpha$ , there exists a unique field (the Reeb vector field) $X_{\alpha }$ on $M$ such that^[3]:

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

.

Related Research Articles

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References

↑ http://people.math.gatech.edu/%7Eetnyre/preprints/papers/phys.pdf ^{[ bare URL PDF ]}
↑ http://www2.im.uj.edu.pl/katedry/K.G/AutumnSchool/Monday.pdf ^{[ bare URL PDF ]}
↑ C. Vizman, "Some Remarks on the Quantomorphism Group" (1997)

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.

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[1] ttp://people.math.gatech.edu/%7Eetnyre/preprints/papers/phys.pdf ^{[ bare URL PDF ]}

[2] ttp://www2.im.uj.edu.pl/katedry/K.G/AutumnSchool/Monday.pdf ^{[ bare URL PDF ]}

[3] C. Vizman, "Some Remarks on the Quantomorphism Group" (1997)

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Reeb vector field

Contents

Definition

See also

Related Research Articles

References