# nLab Klein geometry

Contents

## Theorems

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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

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

• 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.

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

## References

The notion of Klein geometry goes back to

• Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872)

translation by M. W. Haskell, A comparative review of recent researches in geometry , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (retyped pdf, retyped pdf, scan of original)

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)

Last revised on March 3, 2019 at 02:28:09. See the history of this page for a list of all contributions to it.