nLab
Cartan connection

Context

\infty-Chern-Weil theory

Differential cohomology

Contents

Idea

The notion of Cartan connection is a special case of that of connection on a bundle which is more general than the notion of affine connection , but more special than the notion of principal connection , in general:

it is an GG-principal connection subject to the constraint that the connection 1-form linearly identifies each tangent space of the base space with a quotient 𝔤/𝔥\mathfrak{g}/\mathfrak{h} of the Lie algebra 𝔤\mathfrak{g} of GG by a sub-Lie algebra 𝔥\mathfrak{h}.

The fiber of the bundle underlying a Cartan connection is a homogeneous space. This notion is closely related to Klein geometries.

Cartan connections are also just called Cartan geometries .

Definition

Let GG be a Lie group and HGH \hookrightarrow G a sub-Lie group. (So that we may think of the coset space G/HG/H as a Klein geometry.) Write 𝔥𝔤\mathfrak{h} \hookrightarrow \mathfrak{g} for the corresponding Lie algebras.

Definition

A (HG)(H \hookrightarrow G)-Cartan connection over a smooth manifold XX is;

  • a GG-affine connection \nabla on XX;

  • such that

    1. there is a reduction of structure groups along HGH \hookrightarrow G;

    2. for each point xXx \in X the canonical composite (for any local trivialization)

      T xX𝔤𝔤/𝔥 T_x X \stackrel{\nabla}{\to} \mathfrak{g} \to \mathfrak{g}/\mathfrak{h}

      is an isomorphism.

This appears for instance as (Sharpe, section 5.1).

Examples

(pseudo-)Riemannian geometry

Let G=Iso(d,1)G = Iso(d,1) be the Poincare group and HGH \subset G the orthogonal group O(d,1)O(d,1). Then the quotient

𝔦𝔰𝔬(d,1)/𝔰𝔬(d,1) d+1 \mathfrak{iso}(d,1)/\mathfrak{so}(d,1) \simeq \mathbb{R}^{d+1}

is Lorentzian spacetime. Therefore an (O(d,1)Iso(d,1))(O(d,1)\hookrightarrow Iso(d,1))-Cartan connection is equivalently an O(d,1)O(d,1)-connection on a manifold whose tangent spaces look like Minkowski spacetime: this is equivalently a pseudo-Riemannian manifold from the perspective discussed at first-order formulation of gravity:

the d+1\mathbb{R}^{d+1}-valued part of the connection is the vielbein.

local model spaceglobal geometrydifferential cohomologyfirst order formulation of gravity
generalKlein geometryCartan geometryCartan connection
examplesEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
Lorentzian supergeometysupergeometrysuperconnectionsupergravity
generalKlein 2-geometryCartan 2-geometry
higher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

References

Élie Cartan has introduced Cartan connections in his work on the Cartan’s “method of moving frames” (cf. Cartan geometry).

A standard textbook reference is

  • R. Sharpe, Differential Geometry – Cartan’s Generalization of Klein’s Erlagen program Springer (1997)

See also

Revised on November 24, 2014 15:51:44 by Urs Schreiber (89.204.155.187)