Conformal connection
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 |
Comments
Except when stated otherwise, Greek indices run from to
and Latin indices run from
to
in the article above.
For the notion of principal part (of a bundle mapping) cf. the editorial comments to Connections on a manifold.
Conformal connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conformal_connection&oldid=13223