Homogeneous space

Riemannian symmetric spaces are an important class of homogeneous spaces, and include many of the examples listed below.

For example, in the line geometry case, we can identify H as a 12-dimensional subgroup of the 16-dimensional general linear group, GL(4), defined by conditions on the matrix entries

by looking for the stabilizer of the subspace spanned by the first two standard basis vectors. That shows that X has dimension 4.

Since the homogeneous coordinates given by the minors are 6 in number, this means that the latter are not independent of each other. In fact, a single quadratic relation holds between the six minors, as was known to nineteenth-century geometers.

This example was the first known example of a Grassmannian, other than a projective space. There are many further homogeneous spaces of the classical linear groups in common use in mathematics.

The definition is more restrictive than it initially appears: such spaces have remarkable properties, and there is a classification of irreducible prehomogeneous vector spaces, up to a transformation known as "castling".