Conformal connection

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A differential-geometric structure on a smooth manifold , a special form of a connection on a manifold when the smooth fibre bundle with base has as its typical fibre the conformal space of dimension . The structure of attaches to each point a copy of the conformal space , which is identified (up to a conformal transformation preserving and all directions at it) with the tangent space , extended by a point at infinity. The conformal connection as a connection in this space associates with each smooth curve with origin and each point of it, a conformal mapping such that a certain condition is satisfied (see below for the condition on ). Suppose that the space is described by a frame consisting of two points (vertices) and mutually-orthogonal hypersurfaces passing through them. Such a frame is interpreted in the pseudo-Euclidean space as an equivalence class of bases satisfying the conditions

 (1)

with respect to the equivalence

Suppose that is covered by coordinate regions and that in each domain a smooth field of frames in is fixed, such that the vertex defined by the vector is the same as . The condition on is as follows: As , when is displaced along towards , must converge to the identity mapping, and the principal part of its deviation from the latter must be defined, relative to the field of the frame in some neighbourhood of , by a matrix of the form

 (2)

of linear differential forms , , , , of type

 (3)

In other words, the image under of the frame at must be defined by the vectors

where is the tangent vector to at and

Under a transformation of the frame of the field at an arbitrary point according to the formulas , , preserving condition (1), that is, under a passage to an arbitrary element of the principal fibre bundle of conformal frames in the spaces , the forms (3) are replaced by the following -forms on :

that also form a matrix of the form (2). The -forms

form a matrix of the same structure as (2) and are expressed by the formulas in terms of the form , which in view of (3) are linear combinations of the and hence of . For elements of the matrix one has the structure equations of a conformal connection (where for simplicity the primes are omitted):

 (4a)
 (4b)
 (4c)
 (4d)

Here the right-hand sides are semi-basic, that is, they are linear combinations of the only; they form a system of torsion-curvature forms of the conformal connection and are transformed according to the rules

The equations have an invariant sense and determine a conformal connection of zero torsion. Let

Then for :

and for :

The invariant identities , determine the special class of so-called (Cartan) normal conformal connections.

The forms (3), forming a matrix of type (2), uniquely determine the conformal connection on : The image under of the frame at is defined by the solution of the system

with initial conditions , where are the equations of the curve in some coordinate neighbourhood of the point of it with coordinates . Any -forms , , , on satisfying equations (4a)–(4d) with right-hand sides expressed in terms of , where the () are linearly independent, determine a conformal connection on in the above sense.

Conformal connections provide a convenient apparatus for the study of conformal mappings of Riemannian spaces. A conformal connection reduces to the Levi-Civita connection of some Riemannian space if there exists local fields of frames on with respect to which

For the curvature tensor of this connection, defined by the equation

one has

Conversely, for each Levi-Civita connection of a Riemannian space there exists a unique normal conformal connection from which it is obtained in the above way. Here and is expressed in terms of the Ricci tensor and the scalar curvature by the formula

The corresponding tensor is called the conformal curvature tensor of the Levi-Civita connection. Two Riemannian spaces are conformally equivalent if their Levi-Civita connections have the same normal conformal connections. In particular, for , a Riemannian space is conformally Euclidean if and only if for it.

References

 [1] E. Cartan, "Les espaces à connexion conforme" Ann. Soc. Polon. Math. , 2 (1923) pp. 171–221 [2] K. Ogiue, "Theory of conformal connections" Kodai Math. Sem. Reports , 19 (1967) pp. 193–224