# 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 **R**^{n+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
**R**^{n+1,1})

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

## References

- 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

- Hazewinkel, Michiel, ed. (2001), "Conformal connection",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Conformal_connection&oldid=13223

Original source: https://en.wikipedia.org/wiki/Conformal connection.
Read more |