Conformal connection
Encyclopedia
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. Any manifold equipped with a conformal structure has a canonical conformal connection called the normal Cartan connection.
such that the solder form
induced by these data is an isomorphism.
Conformal geometry
In mathematics, conformal geometry is the study of the set of angle-preserving transformations on a space. In two real dimensions, conformal geometry is precisely the geometry of Riemann surfaces...
differential geometry, a conformal connection is a Cartan connection
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 geometry of the principal bundle is tied to the...
on an n-dimensional manifold M arising as a deformation of the Klein geometry
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 the symmetry group of the geometry.For background and motivation...
given by the celestial n-sphere
Celestial sphere
In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...
, viewed as the homogeneous space
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 continuously by symmetry in a transitive way. A special case of this is when the topological group,...
- O+(n+1,1)/P
where P is the stabilizer of a fixed null line through the origin in Rn+2, in the orthochronous Lorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
O+(n+1,1) in n+2 dimensions. 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 a O+(n+1,1)-bundle equipped with- a O+(n+1,1)-connection (the Cartan connectionCartan connectionIn 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 geometry of the principal bundle is tied to the...
) - 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
Solder form
In mathematics, more precisely in differential geometry, a soldering of a fibre bundle to a smooth manifold is a manner of attaching the fibres to the manifold in such a way that they can be regarded as tangent...
induced by these data is an isomorphism.