nLab Klein geometry

Idea

The notion of Klein geometry is essentially that of homogeneous space (coset space) $G/H$ in the context of differential geometry. This is named “Klein geometry” due to its central role in Felix Klein‘s Erlangen program, see below at History.

Klein geometries form the local models for Cartan geometries.

For the generalization of Klein geometry to higher category theory see higher Klein geometry.

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

$$X \coloneqq G/H$$

is connected. $G$ acts transitively on the homogeneous space $X$. We may think of $H\hookrightarrow G$ as the stabilizer subgroup of a point in $X$.

See there at Examples – Stabilizers of shapes / Klein geometry.

History

In (Klein 1872) (the “Erlangen program”) is first of all, in section 1, considered the general idea of (what in modern language one would call) the action of a Lie group “of transformations” on a smooth manifold. The group of all such transformations

by which the geometric properties of configurations in space remain entirely unchanged

is called the Hauptgruppe, principal group.

Then in (Klein 1872, end of section 5) it says:

Suppose in space some group or other, the principal group for instance, be given. Let us then select a single configuration, say a point, or a straight line, or even an ellipsoid, etc., and apply to it all the transformations of the principal group. We thus obtain an infinite manifoldness with a number of dimensions in general equal to the number of arbitrary parameters contained in the group, but reducing in special cases, namely, when the configuration originally selected has the property of being transformed into itself by an infinite number of the transformations of the group. Every manifoldness generated in this way may be called, with reference to the generating group, a body.

This means in modern language, that if $G$ is the given group acting on a given space $X$, and if $S \hookrightarrow X$ is a given subspace (a configuration), then the “body” generated by this is the coset $G/Stab_G(S)$ of $G$ by the stabilizer subgroup $Stab_G(X)$ of that configuration. See also there at Stabilizer of shapes – Klein geometry.

The text goes on to argue that spaces of this form $G/Stab_G(S)$ are of fundamental importance:

If now we desire to base our investigations upon the group, selecting at the same time certain definite configurations as space-elements, and if we wish to represent uniformly things which are of like characteristics, we must evidently choose our space-elements in such a way that their manifoldness either is itself a body or can be decomposed into bodies.

Examples

Example

For $G = E(n)$, the Euclidean group in $n$-dimensions; $H = O(n)$, the orthogonal group; then, $X$ is $n$-dimensional Cartesian space.

Example

Analogously, for $G = Iso(d,1)$ the Poincare group of $(d+1)$-dimensional Minkowski space, and $H = O(d,1)$ the Lorentz group, then $X = \mathbb{R}^{d+1}$ is Minkowski space itself.

Passing to the corresponding Cartan geometry – by what physicists call gauging – yields the first order formulation of gravity.

Example

(crystallography)
The study of crystallographic groups in crystallography is much in the spirit of Klein geometry/the Erlangen program (see for instance Weyl 1938; Grünbaum & Shephard 2010; Engel 1986).

Concretely, the quotient space/quotient orbifold of the space of wave vectors/momenta in a crystal lattice by the (dual) crystallographic group is the Brillouin torus(-orbifold), in terms of which much of condensed matter theory is formulated (see for instance the electron energy bands, the valence bundle, and the K-theory classification of topological phases).

Related concepts

geometric context	gauge group	stabilizer subgroup	local model space	local geometry	global geometry	differential cohomology	first order formulation of gravity
differential geometry	Lie group/algebraic group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	Klein geometry	Cartan geometry	Cartan connection
examples	Euclidean group $Iso(d)$	rotation group $O(d)$	Cartesian space $\mathbb{R}^d$	Euclidean geometry	Riemannian geometry	affine connection	Euclidean gravity
	Poincaré group $Iso(d-1,1)$	Lorentz group $O(d-1,1)$	Minkowski spacetime $\mathbb{R}^{d-1,1}$	Lorentzian geometry	pseudo-Riemannian geometry	spin connection	Einstein gravity
	anti de Sitter group $O(d-1,2)$	$O(d-1,1)$	anti de Sitter spacetime $AdS^d$				AdS gravity
	de Sitter group $O(d,1)$	$O(d-1,1)$	de Sitter spacetime $dS^d$				deSitter gravity
	linear algebraic group	parabolic subgroup/Borel subgroup	flag variety	parabolic geometry
	conformal group $O(d,t+1)$	conformal parabolic subgroup	Möbius space $S^{d,t}$		conformal geometry	conformal connection	conformal gravity
supergeometry	super Lie group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	super Klein geometry	super Cartan geometry	Cartan superconnection
examples	super Poincaré group	spin group	super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$	Lorentzian supergeometry	supergeometry	superconnection	supergravity
	super anti de Sitter group		super anti de Sitter spacetime
higher differential geometry	smooth 2-group $G$	2-monomorphism $H \to G$	homotopy quotient $G//H$	Klein 2-geometry	Cartan 2-geometry
	cohesive ∞-group	∞-monomorphism (i.e. any homomorphism) $H \to G$	homotopy quotient $G//H$ of ∞-action	higher Klein geometry	higher Cartan geometry	higher Cartan connection
examples			extended super Minkowski spacetime		extended supergeometry		higher supergravity: type II, heterotic, 11d

• Clifford-Klein space form

• spherical space form

• (d,d) Klein space

References

General

The notion of Klein geometry goes back to

• Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872) Mathematische Annalen volume 43, pages 63–100 1893 (doi:10.1007/BF01446615)

  English translation by M. W. Haskell:

  A comparative review of recent researches in geometry, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (euclid:1183407629, LaTeX version retyped by Nitin C. Rughoonauth: arXiv:0807.3161)

in the context of what came to be known as the Erlangen program.

Review:

• Vladimir Kisil, Erlangen Programme at Large: An Overview (arXiv:1106.1686)

Monograph account in the broader context of Cartan geometry:

• Richard W. Sharpe: Shapes Fantastic: Klein Geometries, Chapter 4 in: Differential geometry – Cartan’s generalization of Klein’s Erlagen program, Graduate Texts in Mathematics 166, Springer (1997) [ISBN:9780387947327]

Alternative terminology

Beware that some authors say “Kleinian geometry” for pseudo-Riemannian geometry of signature $(--++)$:

• John Barrett, Gary W. Gibbons, M. J. Perry, Christopher N. Pope, P. J. Ruback, Kleinian Geometry and the $N=2$ Superstring, Int. J. Mod. Phys. A9 (1994) 1457-1494 [doi:10.1142/S0217751X94000650, arXiv:hep-th/9302073]