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

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

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 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).

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 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

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

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 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.

Reductive geometries

A Klein geometry (G, H) 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.

	Underlying space	Transformation group G	Subgroup H	Invariants
Projective geometry	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
Conformal geometry on the sphere	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 geometry	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 geometry	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
Spherical geometry	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 geometry	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 geometry	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.