Kleinian Geometry
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Kleinian 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 (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

${\displaystyle g.(aH)=(ga)H.}$

Clearly, this action is transitive (take ), 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 ? 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).

## 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:

${\displaystyle H\to G\to G/H.}$

## 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 by

${\displaystyle K=\{k\in G:g^{-1}kg\in H\;\;\forall g\in G\}.}$

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 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. (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 .

### Reductive geometries

A Klein geometry is said to be reductive and G/H a reductive homogeneous space if the Lie algebra ${\displaystyle {\mathfrak {h}}}$ of H has an H-invariant complement in ${\displaystyle {\mathfrak {g}}}$.

## Examples

In the following table, there is a description of the classical geometries, modeled as Klein geometries.

Projective geometry Conformal geometry on the sphere Hyperbolic geometry Elliptic geometry Spherical geometry Underlying space Transformation group G Subgroup H Invariants Real projective space ${\displaystyle \mathbb {R} \mathrm {P} ^{n}}$ Projective group ${\displaystyle \mathrm {PGL} (n+1)}$ A subgroup ${\displaystyle P}$ fixing a flag ${\displaystyle \{0\}\subset V_{1}\subset V_{n}}$ Projective lines, cross-ratio Sphere ${\displaystyle S^{n}}$ Lorentz group of an ${\displaystyle (n+2)}$-dimensional space ${\displaystyle \mathrm {O} (n+1,1)}$ A subgroup ${\displaystyle P}$ fixing a line in the null cone of the Minkowski metric Generalized circles, angles Hyperbolic space ${\displaystyle H(n)}$, modelled e.g. as time-like lines in the Minkowski space ${\displaystyle \mathbb {R} ^{1,n}}$ Orthochronous Lorentz group ${\displaystyle \mathrm {O} (1,n)/\mathrm {O} (1)}$ ${\displaystyle \mathrm {O} (1)\times \mathrm {O} (n)}$ Lines, circles, distances, angles Elliptic space, modelled e.g. as the lines through the origin in Euclidean space ${\displaystyle \mathbb {R} ^{n+1}}$ ${\displaystyle \mathrm {O} (n+1)/\mathrm {O} (1)}$ ${\displaystyle \mathrm {O} (n)/\mathrm {O} (1)}$ Lines, circles, distances, angles Sphere ${\displaystyle S^{n}}$ Orthogonal group ${\displaystyle \mathrm {O} (n+1)}$ Orthogonal group ${\displaystyle \mathrm {O} (n)}$ Lines (great circles), circles, distances of points, angles Affine space ${\displaystyle A(n)\simeq \mathbb {R} ^{n}}$ Affine group ${\displaystyle \mathrm {Aff} (n)\simeq \mathbb {R} ^{n}\rtimes \mathrm {GL} (n)}$ General linear group ${\displaystyle \mathrm {GL} (n)}$ Lines, quotient of surface areas of geometric shapes, center of mass of triangles Euclidean space ${\displaystyle E(n)}$ Euclidean group ${\displaystyle \mathrm {Euc} (n)\simeq \mathbb {R} ^{n}\rtimes \mathrm {O} (n)}$ Orthogonal group ${\displaystyle \mathrm {O} (n)}$ Distances of points, angles of vectors, areas

## References

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