In
conformal differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, 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, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
: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
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 physicis ...
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 physicis ...
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) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two d ...
has a canonical conformal connection called the normal Cartan connection.
Formal definition
A conformal connection on an ''n''-manifold ''M'' is a
Cartan geometry
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 ...
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
In differential geometry, a ''G''-structure on an ''n''- manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''.
The notion of ''G''-structures includes var ...
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. Intuitiv ...
induced by these data is an isomorphism.
References
*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=
Ülo Lumiste
Conformal geometry
Connection (mathematics)