nLab
Cartan connection

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 $G$-principal connection subject to the constraint that the connection 1-form linearly identifies each tangent space of the base space with a quotient $𝔤/𝔥$ of the Lie algebra $𝔤$ of $G$ by a sub-Lie algebra $𝔥$.

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 $G$ be a Lie group and $H\hookrightarrow G$ a sub-Lie group. (So that we may think of the coset space $G/H$ as a Klein geometry.) Write $𝔥\hookrightarrow 𝔤$ for the corresponding Lie algebras.

Definition

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

• a $G$-affine connection $\nabla$ on $X$;

• such that

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

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

     $$T_{x}X\stackrel{\nabla }{\to }𝔤\to 𝔤/𝔥$$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 $G=\mathrm{Iso}(d,1)$ be the Poincare group and $H\subset G$ the orthogonal group $O(d,1)$. Then the quotient

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

is Lorentzian spacetime. Therefore an $(O(d,1)\hookrightarrow \mathrm{Iso}(d,1))$-Cartan connection is equivalently an $O(d,1)$-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 ${ℝ}^{d+1}$-valued part of the connection is the vielbein.

Related concepts

• connection on a bundle

  • parallel transport, holonomy

• principal connection

  • affine connection, Levi-Civita connection, Cartan connection

• connection on a 2-bundle

• connection on an ∞-bundle

  • ∞-Cartan connection

  • higher parallel transport

local model	global geometry
Klein geometry	Cartan geometry
Klein 2-geometry	Cartan 2-geometry
higher Klein geometry	higher Cartan geometry

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

• wikipedia Cartan connection