Cartan connection

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

satisfying, for every $p\in P$:

1. (Linear isomorphism) The map $\omega_p\colon T_pP\to \mathfrak{g}$ is a linear isomorphism.
2. (Equivariance) For all $h\in H$, $(R_h)^*\omega = \mathrm{Ad}(h^{-1})\omega$.
3. (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

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

1. 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 $G/H$.
2. 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 $O(n)$).
3. 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.