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

O+(n+1,1)/P

where P is the stabilizer of a fixed null line through the origin in Rn+2, in the orthochronous Lorentz group O+(n+1,1) 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+(n+1,1). In other words it is an O+(n+1,1)-bundle equipped with

  • a O+(n+1,1)-connection (the Cartan connection)
  • a reduction of structure group to the stabilizer of a point in the conformal sphere (a null line in Rn+1,1)

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

References
  • Le, Anbo. "Cartan connections for CR manifolds." manuscripta mathematica 122.2 (2007): 245-264.
  • Ü. Lumiste (2001) [1994], "Conformal connection", Encyclopedia of Mathematics, EMS Press
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