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A differential-geometric object on a smooth principal fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c0251701.png" /> that is used to define a [[Horizontal distribution|horizontal distribution]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c0251702.png" /> of a connection in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c0251703.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c0251704.png" /> be the bundle of all tangent frames to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c0251705.png" /> such that the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c0251706.png" /> vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c0251707.png" /> are tangent to the corresponding fibre, and are generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c0251708.png" /> basis elements in the Lie algebra of the structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c0251709.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517011.png" />. A connection object then consists of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517013.png" /> such that the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517014.png" /> is spanned by the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517015.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517016.png" />. Furthermore, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517017.png" /> must satisfy the following conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517018.png" />:
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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/c025/c025170/c02517019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|done}}
  
They are expressed by using the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517020.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517021.png" /> that occur in the structure equations for the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517022.png" /> given by the co-basis dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517023.png" />;
+
A differential-geometric object on a smooth principal fibre bundle  $  P $
 +
that is used to define a [[Horizontal distribution|horizontal distribution]]  $  \Delta $
 +
of a connection in  $  P $.
 +
Let  $  R _ {0} ( P) $
 +
be the bundle of all tangent frames to  $  P $
 +
such that the first  $  r $
 +
vectors  $  e _ {1} \dots e _ {r} $
 +
are tangent to the corresponding fibre, and are generated by $  r $
 +
basis elements in the Lie algebra of the structure group  $  G $
 +
of  $  P $,
 +
$  r = \mathop{\rm dim}  G $.  
 +
A connection object then consists of functions  $  \Gamma _ {i}  ^  \rho  $
 +
on $  R _ {0} ( P) $
 +
such that the subspace of  $  \Delta $
 +
is spanned by the vectors  $  e _ {i} + \Gamma _ {i}  ^  \rho  e _  \rho  $
 +
$  ( \rho , \sigma = 1 \dots r;  i , j , \dots = r + 1 \dots r+ n ) $.  
 +
Furthermore, the  $  \Gamma _ {i}  ^  \rho  $
 +
must satisfy the following conditions on  $  R _ {0} ( P) $:
  
<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/c025/c025170/c02517024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
d \Gamma _ {i}  ^  \rho  - \Gamma _ {j}  ^  \rho  \omega _ {i}  ^ {j} +
 +
\Gamma _ {i}  ^  \sigma  \omega _  \sigma  ^  \rho  + \omega _ {i}  ^  \rho
 +
= \Gamma _ {ij}  ^  \rho  \omega  ^ {j} .
 +
$$
  
A connection object also defines a corresponding [[Connection form|connection form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517025.png" />, given by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517026.png" />, and its [[Curvature form|curvature form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517027.png" />, given by the formulas:
+
They are expressed by using the $  1 $-
 +
forms on  $  R _ {0} ( P) $
 +
that occur in the structure equations for the forms  $  \omega  ^ {i} , \omega  ^  \rho  $
 +
given by the co-basis dual to  $  \{ e _ {i} , e _  \rho  \} $;
  
<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/c025/c025170/c02517028.png" /></td> </tr></table>
+
$$ \tag{2 }
 +
\left .
  
<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/c025/c025170/c02517029.png" /></td> </tr></table>
+
\begin{array}{c}
 +
d \omega  ^ {i}  = \omega  ^ {j} \wedge \omega _ {j}  ^ {i} ,  \\
 +
d \omega  ^  \rho  = -
 +
\frac{1}{2}
 +
C _ {\sigma \tau }  ^  \rho
 +
\omega  ^  \sigma  \wedge \omega  ^  \tau  + \omega  ^ {i} \wedge
 +
\omega _ {i}  ^  \rho  ,  \\
 +
\omega _  \sigma  ^  \rho  = - C _ {\sigma \tau }  ^  \rho
 +
\omega  ^  \tau  . \\
 +
\end{array}
 +
\right \}
 +
$$
  
For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517030.png" /> be the space of affine tangent frames of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517031.png" />-dimensional smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517032.png" />. Then the second equation in (2) has the form
+
A connection object also defines a corresponding [[Connection form|connection form]]  $  \theta $,  
 +
given by the relation  $  \theta  ^  \rho  = \omega  ^  \rho  - \Gamma _ {i}  ^  \rho  \omega  ^ {i} $,
 +
and its [[Curvature form|curvature form]]  $  \Omega $,
 +
given by the formulas:
  
<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/c025/c025170/c02517033.png" /></td> </tr></table>
+
$$
 +
\Omega  ^  \rho  = -
 +
\frac{1}{2}
 +
 
 +
R _ {ij}  ^  \rho  \omega  ^ {i} \wedge \omega  ^ {j} ,
 +
$$
 +
 
 +
$$
 +
R _ {ij}  ^  \rho  = - 2 ( \Gamma _ {[ ij ] }
 +
^  \rho  + C _ {\sigma \tau }  ^  \rho  \Gamma _ {i}  ^  \sigma  \Gamma _ {j}  ^  \tau  ) .
 +
$$
 +
 
 +
For example, let  $  P $
 +
be the space of affine tangent frames of an  $  n $-
 +
dimensional smooth manifold  $  M $.  
 +
Then the second equation in (2) has the form
 +
 
 +
$$
 +
d \omega _ {j}  ^ {i}  = \
 +
- \omega _ {k}  ^ {i} \wedge \omega _ {j}  ^ {k} +
 +
\omega  ^ {k} \wedge \omega _ {jk}  ^ {i}
 +
$$
  
 
and (1) reduces to
 
and (1) reduces to
  
<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/c025/c025170/c02517034.png" /></td> </tr></table>
+
$$
 +
d \Gamma _ {ik}  ^ {j} - \Gamma _ {lk}  ^ {j} \omega _ {i}  ^ {l} -
 +
\Gamma _ {il}  ^ {j} \omega _ {k}  ^ {l} + \Gamma _ {ik}  ^ {l}
 +
\omega _ {l}  ^ {j} + \omega _ {ik}  ^ {j}  = \
 +
\Gamma _ {jkl}  ^ {i} \omega  ^ {l} .
 +
$$
 +
 
 +
Under [[Parallel displacement(2)|parallel displacement]] one must have  $  \omega _ {j}  ^ {i} - \Gamma _ {jk}  ^ {i} \omega  ^ {k} = 0 $.
 +
If a local chart is chosen in  $  M $,
 +
and if in its domain one makes the transition to the natural frame of the chart, i.e. $  \omega  ^ {k} = dx  ^ {k} $,
 +
then the parallel displacement is defined by  $  \omega _ {j}  ^ {i} = \Gamma _ {jk}  ^ {i}  dx  ^ {k} $.  
 +
The classical definition of a connection object of an affine connection on  $  M $
 +
is given by the set of functions  $  \Gamma _ {jk}  ^ {i} $
 +
defined on the domains of the charts such that under transition to the coordinates of another chart these functions are transformed according to the formulas
 +
 
 +
$$
 +
\Gamma _ {st} ^ { \prime r }  = \
 +
 
 +
\frac{\partial  x ^ {\prime r } }{\partial  x  ^ {i} }
 +
 
 +
\frac{\partial  x  ^ {j} }{\partial  x ^ {\prime s } }
 +
 
 +
\frac{\partial  x  ^ {k} }{\partial  x ^ {\prime t } }
 +
 
 +
\Gamma _ {jk}  ^ {i} +
  
Under [[Parallel displacement(2)|parallel displacement]] one must have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517035.png" />. If a local chart is chosen in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517036.png" />, and if in its domain one makes the transition to the natural frame of the chart, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517037.png" />, then the parallel displacement is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517038.png" />. The classical definition of a connection object of an affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517039.png" /> is given by the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025170/c02517040.png" /> defined on the domains of the charts such that under transition to the coordinates of another chart these functions are transformed according to the formulas
+
\frac{\partial  ^ {2} x  ^ {i} }{\partial  x ^ {\prime s } \partial  x ^ {\prime t } }
  
<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/c025/c025170/c02517041.png" /></td> </tr></table>
+
\frac{\partial  x ^ {\prime r } }{\partial  x  ^ {i} }
 +
.
 +
$$
  
 
Here this follows from the condition of invariance under displacement.
 
Here this follows from the condition of invariance under displacement.

Latest revision as of 17:46, 4 June 2020


A differential-geometric object on a smooth principal fibre bundle $ P $ that is used to define a horizontal distribution $ \Delta $ of a connection in $ P $. Let $ R _ {0} ( P) $ be the bundle of all tangent frames to $ P $ such that the first $ r $ vectors $ e _ {1} \dots e _ {r} $ are tangent to the corresponding fibre, and are generated by $ r $ basis elements in the Lie algebra of the structure group $ G $ of $ P $, $ r = \mathop{\rm dim} G $. A connection object then consists of functions $ \Gamma _ {i} ^ \rho $ on $ R _ {0} ( P) $ such that the subspace of $ \Delta $ is spanned by the vectors $ e _ {i} + \Gamma _ {i} ^ \rho e _ \rho $ $ ( \rho , \sigma = 1 \dots r; i , j , \dots = r + 1 \dots r+ n ) $. Furthermore, the $ \Gamma _ {i} ^ \rho $ must satisfy the following conditions on $ R _ {0} ( P) $:

$$ \tag{1 } d \Gamma _ {i} ^ \rho - \Gamma _ {j} ^ \rho \omega _ {i} ^ {j} + \Gamma _ {i} ^ \sigma \omega _ \sigma ^ \rho + \omega _ {i} ^ \rho = \Gamma _ {ij} ^ \rho \omega ^ {j} . $$

They are expressed by using the $ 1 $- forms on $ R _ {0} ( P) $ that occur in the structure equations for the forms $ \omega ^ {i} , \omega ^ \rho $ given by the co-basis dual to $ \{ e _ {i} , e _ \rho \} $;

$$ \tag{2 } \left . \begin{array}{c} d \omega ^ {i} = \omega ^ {j} \wedge \omega _ {j} ^ {i} , \\ d \omega ^ \rho = - \frac{1}{2} C _ {\sigma \tau } ^ \rho \omega ^ \sigma \wedge \omega ^ \tau + \omega ^ {i} \wedge \omega _ {i} ^ \rho , \\ \omega _ \sigma ^ \rho = - C _ {\sigma \tau } ^ \rho \omega ^ \tau . \\ \end{array} \right \} $$

A connection object also defines a corresponding connection form $ \theta $, given by the relation $ \theta ^ \rho = \omega ^ \rho - \Gamma _ {i} ^ \rho \omega ^ {i} $, and its curvature form $ \Omega $, given by the formulas:

$$ \Omega ^ \rho = - \frac{1}{2} R _ {ij} ^ \rho \omega ^ {i} \wedge \omega ^ {j} , $$

$$ R _ {ij} ^ \rho = - 2 ( \Gamma _ {[ ij ] } ^ \rho + C _ {\sigma \tau } ^ \rho \Gamma _ {i} ^ \sigma \Gamma _ {j} ^ \tau ) . $$

For example, let $ P $ be the space of affine tangent frames of an $ n $- dimensional smooth manifold $ M $. Then the second equation in (2) has the form

$$ d \omega _ {j} ^ {i} = \ - \omega _ {k} ^ {i} \wedge \omega _ {j} ^ {k} + \omega ^ {k} \wedge \omega _ {jk} ^ {i} $$

and (1) reduces to

$$ d \Gamma _ {ik} ^ {j} - \Gamma _ {lk} ^ {j} \omega _ {i} ^ {l} - \Gamma _ {il} ^ {j} \omega _ {k} ^ {l} + \Gamma _ {ik} ^ {l} \omega _ {l} ^ {j} + \omega _ {ik} ^ {j} = \ \Gamma _ {jkl} ^ {i} \omega ^ {l} . $$

Under parallel displacement one must have $ \omega _ {j} ^ {i} - \Gamma _ {jk} ^ {i} \omega ^ {k} = 0 $. If a local chart is chosen in $ M $, and if in its domain one makes the transition to the natural frame of the chart, i.e. $ \omega ^ {k} = dx ^ {k} $, then the parallel displacement is defined by $ \omega _ {j} ^ {i} = \Gamma _ {jk} ^ {i} dx ^ {k} $. The classical definition of a connection object of an affine connection on $ M $ is given by the set of functions $ \Gamma _ {jk} ^ {i} $ defined on the domains of the charts such that under transition to the coordinates of another chart these functions are transformed according to the formulas

$$ \Gamma _ {st} ^ { \prime r } = \ \frac{\partial x ^ {\prime r } }{\partial x ^ {i} } \frac{\partial x ^ {j} }{\partial x ^ {\prime s } } \frac{\partial x ^ {k} }{\partial x ^ {\prime t } } \Gamma _ {jk} ^ {i} + \frac{\partial ^ {2} x ^ {i} }{\partial x ^ {\prime s } \partial x ^ {\prime t } } \frac{\partial x ^ {\prime r } }{\partial x ^ {i} } . $$

Here this follows from the condition of invariance under displacement.

How to Cite This Entry:
Connection object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connection_object&oldid=13997
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article