# Isotropic submanifold

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=50362