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 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 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
g ⋅ (aH)=(ga)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 (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:
H\toG\toG/H.
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 by
K=\{k\inG:g-1kg\inH \forallg\inG\}.
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 .
A Klein geometry 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 is a fibration).
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 .
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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 RPn | Projective group PGL(n+1) | A subgroup P \{0\}\subsetV1\subsetVn | |
|---|---|---|---|---|
| Conformal geometry on the sphere | Sphere Sn | Lorentz group of an (n+2) O(n+1,1) | A subgroup P | Generalized circles, angles |
| Hyperbolic geometry | Hyperbolic space H(n) \R1,n | Orthochronous Lorentz group O(1,n)/O(1) | O(1) x O(n) | Lines, circles, distances, angles |
| Elliptic geometry | Elliptic space, modelled e.g. as the lines through the origin in Euclidean space Rn+1 | O(n+1)/O(1) | O(n)/O(1) | Lines, circles, distances, angles |
| Spherical geometry | Sphere Sn | Orthogonal group O(n+1) | Orthogonal group O(n) | Lines (great circles), circles, distances of points, angles |
| Affine geometry | Affine space A(n)\simeq\Rn | Affine group Aff(n)\simeq\Rn\rtimesGL(n) | General linear group GL(n) | Lines, quotient of surface areas of geometric shapes, center of mass of triangles |
| Euclidean geometry | Euclidean space E(n) | Euclidean group Euc(n)\simeq\Rn\rtimesO(n) | Orthogonal group O(n) | Distances of points, angles of vectors, areas |