In
conformal differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, a conformal connection is a
Cartan connection
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the ...
on an ''n''-dimensional manifold ''M'' arising as a deformation of the
Klein geometry
In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space ''X'' together with a transitive action on ''X'' by a Lie group ''G'', which acts as ...
given by the
celestial ''n''-sphere, viewed as the
homogeneous space
In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
: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
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physi ...
O
+(n+1,1) in ''n''+2 dimensions.
Normal Cartan connection
Any manifold equipped with a
conformal structure
In mathematics, conformal geometry is the study of the set of angle-preserving (conformal map, conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space highe ...
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
In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuiti ...
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
*{{springer, id=Conformal_connection&oldid=13223, title=Conformal connection, author=
Ãœlo Lumiste
Conformal geometry
Connection (mathematics)