Klein geometry
# 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 (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

    $g\cdot (aH)=(ga)H.$

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 $x_0$ in X and letting H be the stabilizer subgroup of $x_0$ in G. The group H is necessarily a closed subgroup of G and X is naturally diffeomorphic to G/H.

Two Klein geometries $(G_1,$ $H_1)$ and $(G_2,$ $H_2)$ are geometrically isomorphic if there is a Lie group isomorphism φ : $G_1$ → $G_2$ so that $φ(H_1)$ = $H_2.$ 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 $x_0).$

## 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:

    $H\to G\to G/H.$

## Types of Klein geometries

### Effective 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

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

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 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 $(G_0,$ $G_0$ ∩ H) where $G_0$ is the identity component of G. Note that G = $G_0$ H.

### Reductive geometries

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

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