nLab isotropic subspace

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.

Related concepts

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)