# 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

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.