Klein 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.
Formal definition
A Klein geometry is a pair where G is a Lie group and H is a closed Lie subgroup of G such that the coset space G/H is connected. The group G is called the principal group of the geometry and G/H is called the space of the geometry. The space of a Klein geometry is a smooth manifold of dimensionThere is a natural smooth left action of G on X given by
Clearly, this action is transitive, so that one may then regard X as a homogeneous space for the action of G. The stabilizer of the identity coset is precisely the group H.
Given any connected smooth manifold X and a smooth transitive action by a Lie group G on X, we can construct an associated Klein geometry by fixing a basepoint x0 in X and letting H be the stabilizer subgroup of x0 in G. The group H is necessarily a closed subgroup of G and X is naturally diffeomorphic to G/H.
Two Klein geometries and are geometrically isomorphic if there is a Lie group isomorphism so that. In particular, if φ is conjugation by an element, we see that and are isomorphic. The Klein geometry associated to a homogeneous space X is then unique up to isomorphism.
Bundle description
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:Types of Klein geometries
Effective geometries
The action of G on 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 byThe kernel K may also be described as the core of H 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 and locally effective if K is discrete. If is a Klein geometry with kernel K, then is an effective Klein geometry canonically associated to.
Geometrically oriented geometries
A Klein geometry is geometrically oriented if G is connected.. If H is connected it follows that G is also connected.Given any Klein geometry, there is a geometrically oriented geometry canonically associated to with the same base space G/H. This is the geometry where G0 is the identity component of G. Note that.
Reductive geometries
A Klein geometry is said to be reductive and G/H a reductive homogeneous space if the Lie algebra of H has an H-invariant complement in.Examples
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, 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 |