Cartan connection - Knowlpedia

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ , and let $H\subset G$ be a closed subgroup with Lie algebra $\mathfrak{h}$ .

Definition (Cartan connection)

Let $\pi\colon P\to M$ be a principal $H$ -bundle over a smooth manifold $M$ . A Cartan connection on $P$ (modeled on $(G,H)$ ) is a $\mathfrak{g}$ -valued 1-form

\omega \in \Omega^1(P;\mathfrak{g})

satisfying, for every $p\in P$ :

(Linear isomorphism) The map $\omega_p\colon T_pP\to \mathfrak{g}$ is a linear isomorphism.
(Equivariance) For all $h\in H$ , $(R_h)^*\omega = \mathrm{Ad}(h^{-1})\omega$ .
(Reproduces generators) For every $X\in \mathfrak{h}$ , if $X^\#$ is the corresponding fundamental vertical vector field on $P$ , then $\omega(X^\#)=X$ .

The curvature of a Cartan connection is the $\mathfrak{g}$ -valued 2-form

\Omega := d\omega + \tfrac12[\omega,\omega],

where the bracket uses the Lie bracket on $\mathfrak{g}$ . This generalizes the curvature of a principal connection but with the key strengthening that $\omega$ identifies each tangent space $T_pP$ with $\mathfrak{g}$ .

Examples

Homogeneous model geometry. On the principal $H$ -bundle $G\to G/H$ , the Maurer–Cartan form is a Cartan connection with zero curvature; this is the “flat” Cartan geometry modeled on $G/H$ .
Riemannian geometry as Cartan geometry. On the orthonormal frame bundle of a Riemannian manifold, combining the Levi–Civita connection 1-form with the solder form produces a Cartan connection modeled on Euclidean space (with model group the Euclidean group and stabilizer $O(n)$ ).
Projective and conformal geometries. Standard projective or conformal structures can be encoded as Cartan geometries modeled on appropriate homogeneous spaces; the Cartan curvature measures deviation from the flat model.