Klein geometry

Type of geometry
In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the geometry. For background and motivation see the article on the Erlangen program.

Given a Lie group G and closed subgroup H, there is natural right action of H on G given by right multiplication. This action is both free and proper. The orbits are simply the left cosets of H in G. One concludes that G has the structure of a smooth principal H bundle over the left coset space G/H:

The action of G on 1=X = G/H need not be effective. The kernel of a Klein geometry is defined to be the kernel of the action of G on X. It is given by

The kernel K may also be described as the core of H in G (i. e. the largest subgroup of H that is normal in G). It is the group generated by all the normal subgroups of G that lie in H.

A Klein geometry is said to be effective if 1=K = 1 and locally effective if K is discrete. If (G, H) is a Klein geometry with kernel K, then (G/K, H/K) is an effective Klein geometry canonically associated to (G, H).

A Klein geometry (G, H) is geometrically oriented if G is connected. (This does not imply that G/H is an oriented manifold). If H is connected it follows that G is also connected (this is because G/H is assumed to be connected, and G → G/H is a fibration). Given any Klein geometry (G, H), there is a geometrically oriented geometry canonically associated to (G, H) with the same base space G/H. This is the geometry (G0, G0 ∩ H) where G0 is the identity component of G. Note that 1=G = G0 H.

A Klein geometry (G, H) is said to be reductive and G/H a reductive homogeneous space if the Lie algebra of H has an H invariant complement in .

In the following table, there is a description of the classical geometries, modeled as Klein geometries. Underlying space Transformation group G Subgroup H Invariants Projective geometry Real projective space Projective group A subgroup fixing a flag Projective lines, cross ratio Conformal geometry on the sphere Sphere Lorentz group of an dimensional space A subgroup fixing a line in the null cone of the Minkowski metric Generalized circles, angles Hyperbolic geometry Hyperbolic space , modelled e. g. as time like lines in the Minkowski space Orthochronous Lorentz group Lines, circles, distances, angles Elliptic geometry Elliptic space, modelled e. g. as the lines through the origin in Euclidean space Lines, circles, distances, angles Spherical geometry Sphere Orthogonal group Orthogonal group Lines (great circles), circles, distances of points, angles Affine geometry Affine space Affine group General linear group Lines, quotient of surface areas of geometric shapes, center of mass of triangles Euclidean geometry Euclidean space Euclidean group Orthogonal group Distances of points, angles of vectors, areas