| Homogeneous Space |
Article Index for Homogeneous |
Website Links For Space |
Information AboutHomogeneous Space |
| CATEGORIES ABOUT HOMOGENEOUS SPACE | |
| geometry | |
| topological groups | |
| lie groups | |
| homogeneous spaces | |
|
From the point of view of the Erlangen Programme , one may understand that "all points are the same", in the Geometry of X. This was true of essentially all geometries proposed before Riemannian Geometry . Thus, for example, Euclidean Space , Affine Space and Projective Space are all in natural ways homogeneous spaces for their respective Symmetry Group s. The same is true of the models found of Non-Euclidean Geometry , of constant Curvature , such as Hyperbolic Space . A further classical example is the space of lines in projective space of three dimensions (equivalently, the space of two-dimensional subspaces of a four-dimensional of the 2×4 matrix with columns two basis vectors for the subspace. The geometry of the resulting homogeneous space is the Line Geometry of Julius Plücker . In general, if X is a homogeneous space, and H is the . For example in the line geometry example we can identify H as a 12-dimensional subgroup of the 16-dimensional group GL4, defined by conditions on the matrix entries h13 = h14 = h23 = h24 = 0, 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 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 idea of a Prehomogeneous Vector Space was introduced by Mikio Sato . It is a finite-dimensional Vector Space V with a Group Action of an Algebraic Group G, such that there is an orbit of G that is open for the Zariski Topology (and so, dense). An example is GL1 acting on a one-dimensional space. The definition is more restrictive than it initially appears: such spaces have remarkable properties, and there is a classification. SEE ALSO |
|
|