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.
A Klein geometry is a pair (G, H) where G is a Lie group and H is a closed Lie subgroup of G such that the (left) 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 (or, by an abuse of terminology, simply the Klein geometry). The space X = G/H of a Klein geometry is a smooth manifold of dimension
- dim X = dim G − dim H.
There is a natural smooth left action of G on X given by
Clearly, this action is transitive (take a = 1), so that one may then regard X as a homogeneous space for the action of G. The stabilizer of the identity coset H ∈ X 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 (G, H) 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 (G1, H1) and (G2, H2) are geometrically isomorphic if there is a Lie group isomorphism φ : G1 → G2 so that φ(H1) = H2. In particular, if φ is conjugation by an element g ∈ G, we see that (G, H) and (G, gHg−1) are isomorphic. The Klein geometry associated to a homogeneous space X is then unique up to isomorphism (i.e. it is independent of the chosen basepoint x0).
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
The action of G on 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 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).
Geometrically oriented geometries
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 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|