Conformal connection

In conformal differential geometry, a conformal connection is a Cartan connection on an n-dimensional manifold M arising as a deformation of the Klein geometry given by the celestial n-sphere, viewed as the homogeneous space
where P is the stabilizer of a fixed null line through the origin in Rⁿ⁺², in the orthochronous Lorentz group O⁺ in n+2 dimensions.

Normal Cartan connection

Any manifold equipped with a conformal structure has a canonical conformal connection called the normal Cartan connection.

Formal definition

A conformal connection on an n-manifold M is a Cartan geometry modelled on the conformal sphere, where the latter is viewed as a homogeneous space for O⁺. In other words, it is an O⁺-bundle equipped with

a O⁺-connection
a reduction of structure group to the stabilizer of a point in the conformal sphere

such that the solder form induced by these data is an isomorphism.