Klein geometry

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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[edit]

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 HX 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 : G1G2 so that φ(H1) = H2. In particular, if φ is conjugation by an element gG, 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).

Bundle description[edit]

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[edit]

Effective geometries[edit]

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[edit]

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 GG/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, G0H) where G0 is the identity component of G. Note that G = G0 H.

Reductive geometries[edit]

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 .

Examples[edit]

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

References[edit]

  • R. W. Sharpe (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag. ISBN 0-387-94732-9.