Summary
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 edit
Any manifold equipped with a conformal structure has a canonical conformal connection called the normal Cartan connection.
Formal definition edit
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 edit
- E. Cartan, "Les espaces à connexion conforme", Ann. Soc. Polon. Math., 2 (1923): 171–221.
- K. Ogiue, "Theory of conformal connections" Kodai Math. Sem. Reports, 19 (1967): 193–224.
- Le, Anbo. "Cartan connections for CR manifolds." manuscripta mathematica 122.2 (2007): 245–264.
External links edit
- Ülo Lumiste (2001) [1994], "Conformal connection", Encyclopedia of Mathematics, EMS Press