nLab
isotropic subspace

Contents

Definition

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 WW of inner product spacecondition on orthogonal space W W^\perp
isotropic subspaceWW W \subset W^\perp
coisotropic subspaceW WW^\perp \subset W
Lagrangian subspaceW=W W = W^\perp(for symplectic form)
symplectic spaceWW ={0}W \cap W^\perp = \{0\}(for symplectic form)

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