# nLab isotropic subspace

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# 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 $W$ of inner product spacecondition 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)