In
conformal
Conformal may refer to:
* Conformal (software), in ASIC Software
* Conformal coating in electronics
* Conformal cooling channel, in injection or blow moulding
* Conformal field theory in physics, such as:
** Boundary conformal field theory ...
differential geometry, a conformal connection is a
Cartan connection 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 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 phy ...
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
*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)