Cartan connection

Definition

Let $G$ be a Lie group and $H$ a closed subgroup of $G$ , with Lie algebras $\mathfrak{h} \subset \mathfrak{g}$ . Let $M$ be a smooth manifold with $\dim M = \dim G - \dim H$ . A Cartan connection on $M$ modelled on $(G, H)$ is a principal $H$ -bundle $\pi : P \to M$ together with a smooth $\mathfrak{g}$ -valued $1$ -form $\omega$ on $P$ such that:

$\omega_u : T_uP \to \mathfrak{g}$ is a linear isomorphism for every $u \in P$ .
$R_h^* \omega = \mathrm{Ad}(h^{-1}) \, \omega$ for every $h \in H$ , where $\mathrm{Ad}$ is the adjoint representation of $G$ restricted to $H$ .
$\omega(X^\#) = X$ for every $X \in \mathfrak{h}$ , where $X^\#$ is the fundamental vector field of $X$ on $P$ .

Remarks

The curvature of $\omega$ is $\Omega = d\omega + \frac12 [\omega \wedge \omega]$ . Example: the model itself, $P = G \to G/H$ with the left Maurer–Cartan form $\omega = \theta$ of $G$ . Clause 1 holds because $\theta_g = dL_{g^{-1}}$ is an isomorphism $T_gG \to \mathfrak{g}$ , and clauses 2 and 3 are standard properties of $\theta$ . Its curvature vanishes by the Maurer–Cartan equation, and a Cartan connection is locally isomorphic to this model exactly when $\Omega = 0$ . Non-example: a principal connection form on $P$ has values in $\mathfrak{h}$ , so it fails clause 1 unless $\mathfrak{h} = \mathfrak{g}$ and $M$ is a point. For $G$ the Euclidean group $\R^n \rtimes \mathrm{O}(n)$ and $H = \mathrm{O}(n)$ , a torsion-free Cartan connection is the same as a Riemannian metric.

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Principal bundle concept
Cartan connection definition

Tag	`CTQ0V4WKHN`
URI	`m2://CTQ0V4WKHN`
Kind	definition
Writing	W0 stub
MSC	53C05, 53B15
Defines	Cartan connection
Uses	Principal bundle
Changed	2 November 2026

Cartan connection

Definition

Remarks

Reading path

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