https://books.google.com/books?id=Ytqs4xU5QKAC&lpg=PA178&dq=cartan%2520geometry%2520sharpe&pg=PP1 > v=onepage&q=cartan%2520geometry%2520sharpe&f=false
https://en.wikipedia.org/wiki/Cartan_connection
A Klein geometry consists of a Lie group \(G\) together with a Lie subgroup \(H\) of \(G\). Together \(G\) and \(H\) determine a homogeneous space \(G/H\), on which the group \(G\) acts by left-translation. Klein’s aim was then to study objects living on the homogeneous space which were congruent by the action of \(G\).
A Cartan geometry extends the notion of a Klein geometry by attaching to each point of a manifold a copy of a Klein geometry, and to regard this copy as tangent to the manifold. Thus the geometry of the manifold is infinitesimally identical to that of the Klein geometry, but globally can be quite different. In particular, Cartan geometries no longer have a well-defined action of \(G\) on them. However, a Cartan connection supplies a way of connecting the infinitesimal model spaces within the manifold by means of parallel transport.