Difference between revisions of "Isotropic submanifold"
From Encyclopedia of Mathematics
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| − | A term used in symplectic and contact geometry. In the case of a [[Symplectic manifold|symplectic manifold]] | + | <!--This article has been texified automatically. Since there was no Nroff source code for this article, |
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| + | A term used in symplectic and contact geometry. In the case of a [[Symplectic manifold|symplectic manifold]] $( M ^ { 2 n } , \omega )$, where $\omega$ is a closed, non-degenerate $2$-form, it denotes a submanifold $L$ of $M$ such that $\omega$ restricts to zero on the tangent bundle of $L$. In the case of a contact manifold $( M ^ { 2 n + 1 } , \xi )$, where locally $\xi = \operatorname{ker} \alpha$ with a $1$-form $\alpha$ satisfying $\alpha \wedge ( d \alpha ) ^ { n } \neq 0$, it refers to a submanifold which is everywhere tangent to $\xi $. In either case an isotropic submanifold is of dimension at most $n$. An isotropic submanifold of maximal dimension $n$ is called a Lagrange submanifold in the former case, and a Legendre submanifold in the latter. | ||
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> V.I. Arnold, A.B. Givental, "Symplectic geometry" V.I. Arnold (ed.) S.P. Novikov (ed.) , ''Dynamical Systems IV'' , ''Encycl. Math. Sci.'' , '''4''' , Springer (1990)</td></tr></table> |
Latest revision as of 17:00, 1 July 2020
A term used in symplectic and contact geometry. In the case of a symplectic manifold $( M ^ { 2 n } , \omega )$, where $\omega$ is a closed, non-degenerate $2$-form, it denotes a submanifold $L$ of $M$ such that $\omega$ restricts to zero on the tangent bundle of $L$. In the case of a contact manifold $( M ^ { 2 n + 1 } , \xi )$, where locally $\xi = \operatorname{ker} \alpha$ with a $1$-form $\alpha$ satisfying $\alpha \wedge ( d \alpha ) ^ { n } \neq 0$, it refers to a submanifold which is everywhere tangent to $\xi $. In either case an isotropic submanifold is of dimension at most $n$. An isotropic submanifold of maximal dimension $n$ is called a Lagrange submanifold in the former case, and a Legendre submanifold in the latter.
References
| [a1] | V.I. Arnold, A.B. Givental, "Symplectic geometry" V.I. Arnold (ed.) S.P. Novikov (ed.) , Dynamical Systems IV , Encycl. Math. Sci. , 4 , Springer (1990) |
How to Cite This Entry:
Isotropic submanifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isotropic_submanifold&oldid=15492
Isotropic submanifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isotropic_submanifold&oldid=15492
This article was adapted from an original article by H. Geiges (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article