 The fibers of formula_5 admit inner products with signature (N,N) given by

A linear subspace of formula_5 in which all pairs of vectors have zero inner product is said to be an isotropic subspace.The fibers of formula_5 are "2N"-dimensional and the maximal dimension of an isotropic subspace is "N".An "N"-dimensional isotropic subspace is called a maximal isotropic subspace.