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 of inner product space | condition on orthogonal space | |
|---|---|---|
| isotropic subspace | ||
| coisotropic subspace | ||
| Lagrangian subspace | (for symplectic form) | |
| symplectic space | (for symplectic form) |