Conformal connection

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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+(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.


  • 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.

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