Conformal connection

A differential-geometric structure on a smooth manifold $ M $, a special form of a connection on a manifold when the smooth fibre bundle $ E $ with base $ M $ has as its typical fibre the conformal space $ C _ {n} $ of dimension $ n = \mathop{\rm dim} M $. The structure of $ E $ attaches to each point $ x \in M $ a copy $ ( C _ {n} ) _ {x} $ of the conformal space $ C _ {n} $, which is identified (up to a conformal transformation preserving $ x $ and all directions at it) with the tangent space $ T _ {x} ( M) $, extended by a point at infinity. The conformal connection as a connection in this space $ E $ associates with each smooth curve $ {\mathcal L} \subset M $ with origin $ x _ {0} $ and each point $ x _ {t} $ of it, a conformal mapping $ \gamma _ {t} : ( C _ {n} ) _ {x _ {t} } \rightarrow ( C _ {n} ) _ {x _ {0} } $ such that a certain condition is satisfied (see below for the condition on $ \gamma _ {t} $). Suppose that the space $ C _ {n} $ is described by a frame consisting of two points (vertices) and $ n $ mutually-orthogonal hypersurfaces passing through them. Such a frame is interpreted in the pseudo-Euclidean space $ {} ^ {1} R _ {n+} 2 $ as an equivalence class of bases satisfying the conditions

$$ \tag{1 } \left . \begin{array}{c} ( e _ {0} , e _ {n+} 1 ) = \ ( e _ {1} , e _ {1} ) = \dots = ( e _ {n} , e _ {n} ) , \\ ( e _ {0} , e _ {0} ) = \ ( e _ {n+} 1 , e _ {n+} 1 ) = \ ( e _ {i} , e _ {j} ) = 0 , \\ i , j = 1 \dots n ,\ \ i \neq j , \end{array} \right \} $$

with respect to the equivalence

$$ \{ e _ \alpha \} \sim \ \{ \lambda e _ \alpha \} ,\ \ \alpha = 0 \dots n + 1 . $$

Suppose that $ M $ is covered by coordinate regions and that in each domain a smooth field of frames in $ ( C _ {n} ) _ {x} $ is fixed, such that the vertex defined by the vector $ e _ {0} $ is the same as $ x $. The condition on $ \gamma _ {t} $ is as follows: As $ t \rightarrow 0 $, when $ x _ {t} $ is displaced along $ {\mathcal L} $ towards $ x _ {0} $, $ \gamma _ {t} $ 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 $ x _ {0} $, by a matrix of the form

$$ \tag{2 } \omega = \| \omega _ \alpha ^ \beta \| = \ \left \| \begin{array}{crc} \omega _ {0} ^ {0} &\omega _ {0} ^ {j} & 0 \\ \omega _ {i} ^ {0} &\omega _ {i} ^ {j} &- \omega _ {0} ^ {i} \\ 0 &- \omega _ {j} ^ {0} &- \omega _ {0} ^ {0} \\ \end{array} \right \| , $$

$$ \omega _ {i} ^ {j} + \omega _ {j} ^ {i} = 0,\ \alpha , \beta = 0 \dots n+ 1; \ i, j = 1 \dots n, $$

of $ ( n + 1 ) ( n + 2 ) / 2 $ linear differential forms $ \omega _ {0} ^ {0} $, $ \omega _ {0} ^ {i} $, $ \omega _ {i} ^ {j} $ $ ( i < j ) $, $ \omega _ {i} ^ {0} $, of type

$$ \tag{3 } \omega _ \alpha ^ \beta = \ \Gamma _ {\alpha i } ^ \beta \ d x ^ {i} ,\ \mathop{\rm det} \ \| \Gamma _ {0i} ^ {j} \| \neq 0 . $$

In other words, the image under $ \gamma _ {t} $ of the frame at $ x _ {t} $ must be defined by the vectors

$$ e _ \beta [ \delta _ \alpha ^ \beta + \omega _ \alpha ^ \beta ( X) t + \epsilon _ \alpha ^ \beta ( t) ] , $$

where $ X $ is the tangent vector to $ {\mathcal L} $ at $ x _ {0} $ and

$$ \lim\limits _ {t \rightarrow 0 } \ \frac{\epsilon _ \alpha ^ \beta ( t) }{t} = 0 . $$

Under a transformation of the frame of the field at an arbitrary point $ x $ according to the formulas $ e _ \alpha ^ \prime = A _ \alpha ^ \beta e _ \beta $, $ e _ \beta = A _ \beta ^ {\prime \alpha } e _ \alpha ^ \prime $, preserving condition (1), that is, under a passage to an arbitrary element of the principal fibre bundle $ \Pi $ of conformal frames in the spaces $ ( C _ {n} ) _ {x} $, the forms (3) are replaced by the following $ 1 $- forms on $ \Pi $:

$$ \omega _ \alpha ^ {\prime \beta } = A _ \gamma ^ {\prime \beta } \ d A _ \alpha ^ \gamma + A _ \alpha ^ \gamma A _ \delta ^ {\prime \beta } \omega _ \gamma ^ \delta , $$

that also form a matrix $ \omega ^ \prime $ of the form (2). The $ 2 $- forms

$$ \Omega _ \alpha ^ {\prime \beta } = d \omega _ \alpha ^ {\prime \beta } + \omega _ \gamma ^ {\prime \beta } \wedge \omega _ \alpha ^ {\prime \gamma } $$

form a matrix $ \Omega ^ \prime = \| \Omega _ \alpha ^ {\prime \beta } \| $ of the same structure as (2) and are expressed by the formulas $ \Omega _ \alpha ^ {\prime \beta } = A _ \alpha ^ \gamma A _ \delta ^ {\prime \beta } \Omega _ \gamma ^ \delta $ in terms of the form $ \Omega _ \alpha ^ \beta = d \omega _ \alpha ^ \beta + \omega _ \gamma ^ \beta \wedge \omega _ \alpha ^ \delta $, which in view of (3) are linear combinations of the $ d x ^ {k} \wedge d x ^ {l} $ and hence of $ \omega _ {0} ^ {k} \wedge \omega _ {0} ^ {l} $. For elements of the matrix $ \omega ^ \prime $ one has the structure equations of a conformal connection (where for simplicity the primes are omitted):

$$ \tag{4a } d \omega _ {0} ^ {0} + \omega _ {i} ^ {0} \wedge \omega _ {0} ^ {i} = \ \Omega _ {0} ^ {0} , $$

$$ \tag{4b } d \omega _ {0} ^ {i} + ( \omega _ {j} ^ {i} - \delta _ {j} ^ {i} \omega _ {0} ^ {0} ) \wedge \omega _ {0} ^ {j} = \Omega _ {0} ^ {i} , $$

$$ \tag{4c } d \omega _ {i} ^ {j} + \omega _ {k} ^ {j} \wedge \omega _ {i} ^ {k} + \omega _ {0} ^ {j} \wedge \omega _ {i} ^ {0} + \omega _ {j} ^ {0} \wedge \omega _ {0} ^ {i} = \ \Omega _ {i} ^ {j} ,\ i < j , $$

$$ \tag{4d } d \omega _ {i} ^ {0} + \omega _ {j} ^ {0} \wedge ( \omega _ {i} ^ {j} - \delta _ {i} ^ {j} \omega _ {0} ^ {0} ) = \Omega _ {i} ^ {0} . $$

Here the right-hand sides are semi-basic, that is, they are linear combinations of the $ \omega _ {0} ^ {k} \wedge \omega _ {0} ^ {l} $ only; they form a system of torsion-curvature forms of the conformal connection and are transformed according to the rules

$$ \Omega _ {0} ^ {\prime 0 } = \ A _ {0} ^ {\prime 0 } ( A _ {0} ^ {\prime 0 } \Omega _ {0} ^ {0} + A _ {i} ^ {\prime 0 } \Omega _ {0} ^ {i} ) , $$

$$ \Omega _ {0} ^ {\prime i } = A _ {0} ^ {0} A _ {j} ^ {\prime i } \Omega _ {0} ^ {j} , $$

$$ \Omega _ {i} ^ {\prime j } = A _ {i} ^ {k} A _ {l} ^ {\prime j } \Omega _ {k} ^ {l} + \Omega _ {0} ^ {k} ( A _ {i} ^ {0} A _ {k} ^ {\prime j } - A _ {i} ^ {k} A _ {n+} 1 ^ {\prime j } ) . $$

The equations $ \Omega _ {0} ^ {i} = 0 $ have an invariant sense and determine a conformal connection of zero torsion. Let

$$ \Omega _ {i} ^ {j} = \ \frac{1}{2} C _ {ikl} ^ {j} \omega _ {0} ^ {k} \wedge \omega _ {0} ^ {l} . $$

Then for $ \Omega _ {0} ^ {i} = 0 $:

$$ C _ {ikl} ^ {\prime j } = \ ( A _ {0} ^ {\prime 0 } ) ^ {2} A _ {i} ^ {p} A _ {q} ^ {\prime j } A _ {k} ^ {r} A _ {l} ^ {s} C _ {prs} ^ {q} , $$

and for $ C _ {ik} = C _ {ikj} ^ {j} $:

$$ C _ {ik} ^ \prime = \ ( A _ {0} ^ {\prime 0 } ) ^ {2} A _ {i} ^ {p} A _ {k} ^ {r} C _ {pr} . $$

The invariant identities $ \Omega _ {0} ^ {i} = \Omega _ {0} ^ {0} = 0 $, $ C _ {ik} = 0 $ 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 $ M $: The image under $ \gamma _ {t} : ( C _ {n} ) _ {x _ {t} } \rightarrow ( C _ {n} ) _ {x _ {0} } $ of the frame at $ x _ {t} $ is defined by the solution $ \{ e _ \alpha ( t) \} $ of the system

$$ d u _ \alpha = \ ( \omega _ \alpha ^ \beta ) _ {x ( t ) } ( \dot{x} ( t) ) u _ \beta $$

with initial conditions $ u _ \alpha ( 0) = e _ \alpha $, where $ x ^ {i} = x ^ {i} ( t) $ are the equations of the curve $ {\mathcal L} $ in some coordinate neighbourhood of the point $ x _ {0} $ of it with coordinates $ x ^ {i} ( 0) $. Any $ 1 $- forms $ \omega _ {0} ^ {0} $, $ \omega _ {0} ^ {i} $, $ \omega _ {i} ^ {j} $ $ ( i < j ) $, $ \omega _ {i} ^ {0} $ on $ \Pi $ satisfying equations (4a)–(4d) with right-hand sides expressed in terms of $ \omega _ {0} ^ {k} \wedge \omega _ {0} ^ {l} $, where the $ \omega _ {0} ^ {i} $( $ i = 1 \dots n $) are linearly independent, determine a conformal connection on $ M $ 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 $ M $ with respect to which

$$ \omega _ {i} ^ {0} = \ P _ {ij} \omega _ {0} ^ {j} ,\ \ \omega _ {0} ^ {0} = \ Q _ {i} \omega _ {0} ^ {i} ,\ \ \Omega _ {0} ^ {i} = \ Q _ {j} \omega _ {0} ^ {i} \wedge \omega _ {0} ^ {j} . $$

For the curvature tensor $ R _ {ikl} ^ {j} $ of this connection, defined by the equation

$$ d \omega _ {i} ^ {j} + \omega _ {k} ^ {j} \wedge \omega _ {i} ^ {k} = \ \frac{1}{2} R _ {ikl} ^ {j} \omega _ {0} ^ {k} \wedge \omega _ {0} ^ {l} , $$

one has

$$ R _ {ikl} ^ {j} = \ \delta _ {l} ^ {j} P _ {ik} - \delta _ {k} ^ {j} P _ {il} - \delta _ {l} ^ {i} P _ {jk} + \delta _ {k} ^ {i} P _ {jl} + C _ {ikl} ^ {j} . $$

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 $ Q _ {j} = 0 $ and $ P _ {ij} $ is expressed in terms of the Ricci tensor $ R _ {ik} = R _ {ikj} ^ {j} $ and the scalar curvature $ R = \sum R _ {ii} $ by the formula

$$ P _ {ij} = \frac{1}{n-} 2 R _ {ij} - \delta _ {j} ^ {i} \frac{R}{2 ( n - 1 ) ( n - 2 ) } . $$

The corresponding tensor $ C _ {ikl} ^ {j} $ 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 $ n > 3 $, a Riemannian space is conformally Euclidean if and only if $ C _ {ikl} ^ {j} = 0 $ 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 $ 0 $ to $ n + 1 $ and Latin indices run from $ 1 $ to $ n + 1 $ in the article above.

For the notion of principal part (of a bundle mapping) cf. the editorial comments to Connections on a manifold.