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A [[Differential-geometric structure|differential-geometric structure]] on a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247301.png" />, a special form of a connection on a manifold when the smooth fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247302.png" /> with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247303.png" /> has as its typical fibre the conformal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247304.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247305.png" />. The structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247306.png" /> attaches to each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247307.png" /> a copy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247308.png" /> of the conformal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247309.png" />, which is identified (up to a conformal transformation preserving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473010.png" /> and all directions at it) with the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473011.png" />, extended by a point at infinity. The conformal connection as a connection in this space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473012.png" /> associates with each smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473013.png" /> with origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473014.png" /> and each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473015.png" /> of it, a conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473016.png" /> such that a certain condition is satisfied (see below for the condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473017.png" />). Suppose that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473018.png" /> is described by a frame consisting of two points (vertices) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473019.png" /> mutually-orthogonal hypersurfaces passing through them. Such a frame is interpreted in the pseudo-Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473020.png" /> as an equivalence class of bases satisfying the conditions
+
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$#C+1 = 120 : ~/encyclopedia/old_files/data/C024/C.0204730 Conformal connection
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
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 +
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 +
 
 +
A [[Differential-geometric structure|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
 
with respect to the equivalence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473022.png" /></td> </tr></table>
+
$$
 +
\{ e _  \alpha  \}  \sim \
 +
\{ \lambda e _  \alpha  \} ,\ \
 +
\alpha = 0 \dots n + 1 .
 +
$$
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473023.png" /> is covered by coordinate regions and that in each domain a smooth field of frames in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473024.png" /> is fixed, such that the vertex defined by the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473025.png" /> is the same as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473026.png" />. The condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473027.png" /> is as follows: As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473028.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473029.png" /> is displaced along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473030.png" /> towards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473032.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473033.png" />, by a matrix of the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\omega  = \| \omega _  \alpha  ^  \beta  \|  = \
 +
\left \|
 +
\begin{array}{crc}
 +
\omega _ {0}  ^ {0}  &\omega _ {0}  ^ {j}  & 0  \\
 +
\omega _ {i}  ^ {0}  &\omega _ {i}  ^ {j}  &- \omega _ {0}  ^ {i}  \\
 +
0  &- \omega _ {j}  ^ {0}  &- \omega _ {0}  ^ {0}  \\
 +
\end{array}
 +
\right \| ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473035.png" /></td> </tr></table>
+
$$
 +
\omega _ {i}  ^ {j} + \omega _ {j}  ^ {i}  = 0,\  \alpha
 +
, \beta = 0 \dots n+ 1; \  i, j = 1 \dots n,
 +
$$
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473036.png" /> linear differential forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473039.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473041.png" />, of type
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\omega _  \alpha  ^  \beta  = \
 +
\Gamma _ {\alpha i }  ^  \beta  \
 +
d x  ^ {i} ,\  \mathop{\rm det} \
 +
\| \Gamma _ {0i}  ^ {j} \|
 +
\neq  0 .
 +
$$
  
In other words, the image under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473043.png" /> of the frame at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473044.png" /> must be defined by the vectors
+
In other words, the image under $  \gamma _ {t} $
 +
of the frame at $  x _ {t} $
 +
must be defined by the vectors
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473045.png" /></td> </tr></table>
+
$$
 +
e _  \beta  [ \delta _  \alpha  ^  \beta  + \omega _  \alpha  ^  \beta  ( X) t + \epsilon _  \alpha  ^  \beta  ( t) ] ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473046.png" /> is the tangent vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473047.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473048.png" /> and
+
where $  X $
 +
is the tangent vector to $  {\mathcal L} $
 +
at $  x _ {0} $
 +
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473049.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \rightarrow 0 } \
  
Under a transformation of the frame of the field at an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473050.png" /> according to the formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473052.png" />, preserving condition (1), that is, under a passage to an arbitrary element of the principal fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473053.png" /> of conformal frames in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473054.png" />, the forms (3) are replaced by the following <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473055.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473056.png" />:
+
\frac{\epsilon _  \alpha  ^  \beta  ( t) }{t}
 +
  = 0 .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473057.png" /></td> </tr></table>
+
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 $:
  
that also form a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473058.png" /> of the form (2). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473059.png" />-forms
+
$$
 +
\omega _  \alpha  ^ {\prime \beta }
 +
= A _  \gamma  ^ {\prime \beta } \
 +
d A _  \alpha  ^  \gamma  +
 +
A _  \alpha  ^  \gamma  A _  \delta  ^ {\prime \beta } \omega _  \gamma  ^  \delta  ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473060.png" /></td> </tr></table>
+
that also form a matrix  $  \omega  ^  \prime  $
 +
of the form (2). The  $  2 $-
 +
forms
  
form a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473061.png" /> of the same structure as (2) and are expressed by the formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473062.png" /> in terms of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473063.png" />, which in view of (3) are linear combinations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473064.png" /> and hence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473065.png" />. For elements of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473066.png" /> one has the structure equations of a conformal connection (where for simplicity the primes are omitted):
+
$$
 +
\Omega _  \alpha  ^ {\prime \beta }
 +
= d \omega _  \alpha  ^ {\prime \beta } +
 +
\omega _  \gamma  ^ {\prime \beta }
 +
\wedge \omega _  \alpha  ^ {\prime \gamma }
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473067.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4a)</td></tr></table>
+
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):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473068.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4b)</td></tr></table>
+
$$ \tag{4a }
 +
d \omega _ {0}  ^ {0} + \omega _ {i}  ^ {0} \wedge \omega _ {0}  ^ {i}  = \
 +
\Omega _ {0}  ^ {0} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473069.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4c)</td></tr></table>
+
$$ \tag{4b }
 +
d \omega _ {0}  ^ {i} + ( \omega _ {j}  ^ {i} - \delta _ {j}  ^ {i} \omega _ {0}  ^ {0} ) \wedge \omega _ {0}  ^ {j}  = \Omega _ {0}  ^ {i} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473070.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4d)</td></tr></table>
+
$$ \tag{4c }
 +
d \omega _ {i}  ^ {j} + \omega _ {k}  ^ {j} \wedge \omega _ {i}  ^ {k} +
 +
\omega _ {0}  ^ {j} \wedge \omega _ {i}  ^ {0} + \omega _ {j}  ^ {0} \wedge \omega _ {0}  ^ {i}  = \
 +
\Omega _ {i}  ^ {j} ,\  i < j ,
 +
$$
  
Here the right-hand sides are semi-basic, that is, they are linear combinations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473071.png" /> only; they form a system of torsion-curvature forms of the conformal connection and are transformed according to the rules
+
$$ \tag{4d }
 +
d \omega _ {i}  ^ {0} + \omega _ {j}  ^ {0} \wedge ( \omega _ {i}  ^ {j} -
 +
\delta _ {i}  ^ {j} \omega _ {0}  ^ {0} )  = \Omega _ {i}  ^ {0} .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473072.png" /></td> </tr></table>
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473073.png" /></td> </tr></table>
+
$$
 +
\Omega _ {0} ^ {\prime 0 }  = \
 +
A _ {0} ^ {\prime 0 } ( A _ {0} ^ {\prime 0 } \Omega _ {0}  ^ {0} +
 +
A _ {i} ^ {\prime 0 } \Omega _ {0}  ^ {i} ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473074.png" /></td> </tr></table>
+
$$
 +
\Omega _ {0} ^ {\prime i }  = A _ {0}  ^ {0} A _ {j} ^ {\prime i } \Omega _ {0}  ^ {j} ,
 +
$$
  
The equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473075.png" /> have an invariant sense and determine a conformal connection of zero torsion. Let
+
$$
 +
\Omega _ {i} ^ {\prime j }  = A _ {i}  ^ {k} A _ {l} ^ {\prime j } \Omega _ {k}  ^ {l} + \Omega _ {0}  ^ {k} ( A _ {i}  ^ {0} A _ {k} ^ {\prime j } - A _ {i}  ^ {k} A _ {n+} 1 ^ {\prime j } ) .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473076.png" /></td> </tr></table>
+
The equations  $  \Omega _ {0}  ^ {i} = 0 $
 +
have an invariant sense and determine a conformal connection of zero torsion. Let
  
Then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473077.png" />:
+
$$
 +
\Omega _ {i}  ^ {j}  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473078.png" /></td> </tr></table>
+
\frac{1}{2}
 +
C _ {ikl}  ^ {j}
 +
\omega _ {0}  ^ {k} \wedge
 +
\omega _ {0}  ^ {l} .
 +
$$
  
and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473079.png" />:
+
Then for $  \Omega _ {0}  ^ {i} = 0 $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473080.png" /></td> </tr></table>
+
$$
 +
C _ {ikl} ^ {\prime j }  = \
 +
( A _ {0} ^ {\prime 0 } )  ^ {2}
 +
A _ {i}  ^ {p} A _ {q} ^
 +
{\prime j } A _ {k}  ^ {r} A _ {l}  ^ {s} C _ {prs}  ^ {q} ,
 +
$$
  
The invariant identities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473082.png" /> determine the special class of so-called (Cartan) normal conformal connections.
+
and for  $  C _ {ik} = C _ {ikj}  ^ {j} $:
  
The forms (3), forming a matrix of type (2), uniquely determine the conformal connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473083.png" />: The image under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473084.png" /> of the frame at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473085.png" /> is defined by the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473086.png" /> of the system
+
$$
 +
C _ {ik}  ^  \prime  = \
 +
( A _ {0} ^ {\prime 0 } ) ^ {2}
 +
A _ {i}  ^ {p} A _ {k}  ^ {r}
 +
C _ {pr} .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473087.png" /></td> </tr></table>
+
The invariant identities  $  \Omega _ {0}  ^ {i} = \Omega _ {0}  ^ {0} = 0 $,
 +
$  C _ {ik} = 0 $
 +
determine the special class of so-called (Cartan) normal conformal connections.
  
with initial conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473088.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473089.png" /> are the equations of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473090.png" /> in some coordinate neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473091.png" /> of it with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473092.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473093.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473096.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473098.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473099.png" /> satisfying equations (4a)(4d) with right-hand sides expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730100.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730101.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730102.png" />) are linearly independent, determine a conformal connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730103.png" /> in the above sense.
+
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
  
Conformal connections provide a convenient apparatus for the study of conformal mappings of Riemannian spaces. A conformal connection reduces to the [[Levi-Civita connection|Levi-Civita connection]] of some Riemannian space if there exists local fields of frames on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730104.png" /> with respect to which
+
$$
 +
d u _  \alpha  = \
 +
( \omega _  \alpha  ^  \beta  ) _ {x ( t ) }  ( \dot{x}
 +
( t) ) u _  \beta  $$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730105.png" /></td> </tr></table>
+
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.
  
For the curvature tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730106.png" /> of this connection, defined by the equation
+
Conformal connections provide a convenient apparatus for the study of conformal mappings of Riemannian spaces. A conformal connection reduces to the [[Levi-Civita connection|Levi-Civita connection]] of some Riemannian space if there exists local fields of frames on  $  M $
 +
with respect to which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730107.png" /></td> </tr></table>
+
$$
 +
\omega _ {i}  ^ {0}  = \
 +
P _ {ij} \omega _ {0}  ^ {j} ,\ \
 +
\omega _ {0}  ^ {0}  = \
 +
Q _ {i} \omega _ {0}  ^ {i} ,\ \
 +
\Omega _ {0}  ^ {i}  = \
 +
Q _ {j} \omega _ {0}  ^ {i} \wedge
 +
\omega _ {0}  ^ {j} .
 +
$$
 +
 
 +
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
 
one has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730108.png" /></td> </tr></table>
+
$$
 +
R _ {ikl}  ^ {j}  = \
 +
\delta _ {l}  ^ {j}
 +
P _ {ik} - \delta _ {k}  ^ {j}
 +
P _ {il} - \delta _ {l}  ^ {i}
 +
P _ {jk} + \delta _ {k}  ^ {i}
 +
P _ {jl} + C _ {ikl}  ^ {j} .
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730110.png" /> is expressed in terms of the Ricci tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730111.png" /> and the scalar curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730112.png" /> by the formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730113.png" /></td> </tr></table>
+
$$
 +
P _ {ij}  =
 +
\frac{1}{n-}
 +
2
 +
R _ {ij} - \delta _ {j}  ^ {i}
  
The corresponding tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730114.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730115.png" />, a Riemannian space is conformally Euclidean if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730116.png" /> for it.
+
\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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cartan,  "Les espaces à connexion conforme"  ''Ann. Soc. Polon. Math.'' , '''2'''  (1923)  pp. 171–221</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Ogiue,  "Theory of conformal connections"  ''Kodai Math. Sem. Reports'' , '''19'''  (1967)  pp. 193–224</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cartan,  "Les espaces à connexion conforme"  ''Ann. Soc. Polon. Math.'' , '''2'''  (1923)  pp. 171–221</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Ogiue,  "Theory of conformal connections"  ''Kodai Math. Sem. Reports'' , '''19'''  (1967)  pp. 193–224</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Except when stated otherwise, Greek indices run from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730117.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730118.png" /> and Latin indices run from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730119.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730120.png" /> in the article above.
+
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|Connections on a manifold]].
 
For the notion of principal part (of a bundle mapping) cf. the editorial comments to [[Connections on a manifold|Connections on a manifold]].

Latest revision as of 17:46, 4 June 2020


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.

How to Cite This Entry:
Conformal connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conformal_connection&oldid=46453
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article