nLab Klein geometry




Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



The notion of Klein geometry is essentially that of homogeneous space (coset space) G/HG/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.


A Klein geometry is a pair (G,H)(G, H) where GG is a Lie group and HH is a closed Lie subgroup of GG such that the (left) coset space

XG/H X \coloneqq G/H

is connected. GG acts transitively on the homogeneous space XX. We may think of HGH\hookrightarrow G as the stabilizer subgroup of a point in XX.

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


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 GG is the given group acting on a given space XX, and if SXS \hookrightarrow X is a given subspace (a configuration), then the “body” generated by this is the coset G/Stab G(S)G/Stab_G(S) of GG by the stabilizer subgroup Stab G(X)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)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.



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


Analogously, for G=Iso(d,1)G = Iso(d,1) the Poincare group of (d+1)(d+1)-dimensional Minkowski space, and H=O(d,1)H = O(d,1) the Lorentz group, then X= d+1X = \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.


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).

geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d



The notion of Klein geometry goes back to

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

A review is for instance in

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

Alternative terminology

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

Last revised on November 8, 2023 at 20:52:57. See the history of this page for a list of all contributions to it.