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 $\mathfrak{g}/\mathfrak{h}$ of the Lie algebra $\mathfrak{g}$ of $G$ by a sub-Lie algebra $\mathfrak{h}$.
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 .
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 $\mathfrak{h} \hookrightarrow \mathfrak{g}$ for the corresponding Lie algebras.
A $(H \hookrightarrow G)$-Cartan connection over a smooth manifold $X$ is;
a $G$-affine connection $\nabla$ on $X$;
such that
there is a reduction of structure groups along $H \hookrightarrow G$;
for each point $x \in X$ the canonical composite (for any local trivialization)
is an isomorphism.
This appears for instance as (Sharpe, section 5.1).
Let $G = Iso(d,1)$ be the Poincare group and $H \subset G$ the orthogonal group $O(d,1)$. Then the quotient
is Lorentzian spacetime. Therefore an $(O(d,1)\hookrightarrow Iso(d,1))$-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 $\mathbb{R}^{d+1}$-valued part of the connection is the vielbein.
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
See also