isotropic subspace

A **totally isotropic subspace** of an inner product space is a sub-vector space on which the bilinear form vanishes.

An **isotropic subspace** of a symplectic vector space is a vector subspace on which the symplectic form vanishes.

A maximal isotropic subspace is called a **lagrangian subspace**.

The space of all isotropic subspaces of a given inner product space is called its **isotropic Grassmannian**.

A submanifold of a symplectic manifold each tangent space of which is isotropic with respect to the ambient symplectic structure is an **isotropic submanifold**.

type of subspace $W$ of inner product space | condition on orthogonal space $W^\perp$ | |
---|---|---|

isotropic subspace | $W \subset W^\perp$ | |

coisotropic subspace | $W^\perp \subset W$ | |

Lagrangian subspace | $W = W^\perp$ | (for symplectic form) |

symplectic space | $W \cap W^\perp = \{0\}$ | (for symplectic form) |

Revised on March 18, 2013 23:40:36
by Urs Schreiber
(89.204.138.142)