Namespaces
Variants
Actions

Difference between revisions of "Connection form"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A linear differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c0251501.png" /> on a principal fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c0251502.png" /> that takes values in the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c0251503.png" /> of the structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c0251504.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c0251505.png" />. It is defined by a certain [[Linear connection|linear connection]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c0251506.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c0251507.png" />, and it determines this connection uniquely. The values of the connection form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c0251508.png" /> in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c0251509.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515011.png" />, are defined as the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515012.png" /> which, under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515014.png" />, generate the second component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515015.png" /> relative to the direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515016.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515017.png" /> is the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515018.png" /> that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515020.png" /> is the horizontal distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515021.png" />. The horizontal distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515022.png" />, and so the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515023.png" />, can be recovered from the connection form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515024.png" /> in the following way.
+
<!--
 +
c0251501.png
 +
$#A+1 = 67 n = 0
 +
$#C+1 = 67 : ~/encyclopedia/old_files/data/C025/C.0205150 Connection form
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
The Cartan–Laptev theorem. For a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515026.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515027.png" /> to be a connection form it is necessary and sufficient that: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515028.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515029.png" />, is the element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515030.png" /> that generates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515031.png" /> under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515033.png" />; and 2) the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515034.png" />-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515035.png" />-form
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/c025150/c02515036.png" /></td> </tr></table>
+
A linear differential form  $  \theta $
 +
on a principal fibre bundle  $  P $
 +
that takes values in the Lie algebra  $  \mathfrak g $
 +
of the structure group  $  G $
 +
of  $  P $.
 +
It is defined by a certain [[Linear connection|linear connection]]  $  \Gamma $
 +
on  $  P $,
 +
and it determines this connection uniquely. The values of the connection form  $  \theta _ {y} ( Y) $
 +
in terms of  $  \Gamma $,
 +
where  $  y \in P $
 +
and  $  Y \in T _ {y} ( P) $,
 +
are defined as the elements of  $  \mathfrak g $
 +
which, under the action of  $  G $
 +
on  $  P $,
 +
generate the second component of  $  Y $
 +
relative to the direct sum  $  T _ {y} ( F) = \Delta _ {y} \oplus T _ {y} ( G _ {y} ) $.  
 +
Here  $  G _ {y} $
 +
is the fibre of  $  P $
 +
that contains  $  y $
 +
and  $  \Delta $
 +
is the horizontal distribution of  $  \Gamma $.  
 +
The horizontal distribution  $  \Delta $,
 +
and so the connection  $  \Gamma $,
 +
can be recovered from the connection form  $  \theta $
 +
in the following way.
  
formed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515037.png" />, is semi-basic, or horizontal, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515038.png" /> if at least one of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515039.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515040.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515041.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515042.png" /> is called the [[Curvature form|curvature form]] of the connection. If a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515043.png" /> is defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515044.png" />, then condition 2) can locally be expressed by the equalities:
+
The Cartan–Laptev theorem. For a form  $  \theta $
 +
on  $  P $
 +
with values in  $  \mathfrak g $
 +
to be a connection form it is necessary and sufficient that: 1)  $  \theta _ {y} ( Y) $,  
 +
for  $  Y \in T _ {y} ( G _ {y} ) $,  
 +
is the element of $  \mathfrak g $
 +
that generates  $  Y $
 +
under the action of $  G $
 +
on  $  P $;
 +
and 2) the $  \mathfrak g $-
 +
valued  $  2 $-
 +
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/c025/c025150/c02515045.png" /></td> </tr></table>
+
$$
 +
\Omega  = d \theta +
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515046.png" /> are certain linearly independent semi-basic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515047.png" />-forms. The necessity of condition 2) was established in this form by E. Cartan [[#References|[1]]]; its sufficiency under the additional assumption of 1) was proved by G.F. Laptev [[#References|[2]]]. The equations
+
\frac{1}{2}
 +
[ \theta , \theta ] ,
 +
$$
  
for the components of the connection form are called the structure equations for the connection in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515048.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515049.png" /> define the curvature object.
+
formed from  $  \theta $,
 +
is semi-basic, or horizontal, that is,  $  \Omega _ {y} ( Y , Y _ {1} ) = 0 $
 +
if at least one of the vectors  $  Y , Y _ {1} $
 +
belongs to  $  T _ {y} ( G _ {y} ) $.
 +
The  $  2 $-
 +
form $  \Omega $
 +
is called the [[Curvature form|curvature form]] of the connection. If a basis  $  \{ e _ {1} \dots e _ {r} \} $
 +
is defined in $  \mathfrak g $,  
 +
then condition 2) can locally be expressed by the equalities:
  
As an example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515050.png" /> be the space of affine frames in the tangent bundle of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515051.png" />-dimensional smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515052.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515054.png" /> are, respectively, the group and the Lie algebra of matrices of the form
+
$$
 +
d \theta  ^  \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/c025150/c02515055.png" /></td> </tr></table>
+
\frac{1}{2}
 +
C _ {\sigma \tau }  ^  \rho
 +
\theta  ^  \sigma  \wedge
 +
\theta  ^  \tau  = \
 +
 
 +
\frac{1}{2}
 +
R _ {ij}  ^  \rho
 +
\omega  ^ {i} \wedge \omega  ^ {j} ,
 +
$$
 +
 
 +
where  $  \omega  ^ {1} \dots \omega  ^ {n} $
 +
are certain linearly independent semi-basic  $  1 $-
 +
forms. The necessity of condition 2) was established in this form by E. Cartan [[#References|[1]]]; its sufficiency under the additional assumption of 1) was proved by G.F. Laptev [[#References|[2]]]. The equations
 +
 
 +
for the components of the connection form are called the structure equations for the connection in  $  P $,
 +
the  $  R _ {ij}  ^  \rho  $
 +
define the curvature object.
 +
 
 +
As an example, let  $  P $
 +
be the space of affine frames in the tangent bundle of an  $  n $-
 +
dimensional smooth manifold  $  M $.  
 +
Then  $  G $
 +
and  $  \mathfrak g $
 +
are, respectively, the group and the Lie algebra of matrices of the form
 +
 
 +
$$
 +
\left \|
 +
\begin{array}{cc}
 +
1  &a  ^ {i}  \\
 +
0  &A _ {j}  ^ {i}  \\
 +
\end{array}
 +
\right \| ,\ \
 +
\mathop{\rm det}  | A _ {j}  ^ {i} |
 +
\neq  0 ,
 +
$$
  
 
and
 
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/c025/c025150/c02515056.png" /></td> </tr></table>
+
$$
 +
\left \|
 +
\begin{array}{cc}
 +
0  &\mathfrak g  ^ {i}  \\
 +
0 &\mathfrak g _ {j}  ^ {i}  \\
 +
\end{array}
 +
\right \| \ \
 +
( i , j = 1 \dots n ) .
 +
$$
 +
 
 +
By the Cartan–Laptev theorem, the  $  \mathfrak g $-
 +
valued  $  1 $-
 +
form
  
By the Cartan–Laptev theorem, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515057.png" />-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515058.png" />-form
+
$$
 +
\theta  = \
 +
\left \|
 +
\begin{array}{cc}
 +
0 &\omega  ^ {i}  \\
 +
0 &\omega _ {j}  ^ {i}  \\
 +
\end{array}
 +
\right \|
 +
$$
  
<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/c025/c025150/c02515059.png" /></td> </tr></table>
+
on  $  P $
 +
is the connection form of a certain [[Affine connection|affine connection]] on  $  M $
 +
if and only if
  
on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515060.png" /> is the connection form of a certain [[Affine connection|affine connection]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515061.png" /> if and only if
+
$$
 +
d \omega  ^ {i} +
 +
\omega _ {j}  ^ {i} \wedge
 +
\omega  ^ {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/c025/c025150/c02515062.png" /></td> </tr></table>
+
\frac{1}{2}
 +
T _ {jk} ^ { i }
 +
\omega  ^ {j} \wedge \omega  ^ {k} ,
 +
$$
  
<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/c025150/c02515063.png" /></td> </tr></table>
+
$$
 +
d \omega _ {j}  ^ {i}  = \omega _ {k}  ^ {i} \wedge \omega _ {j}  ^ {k}  =
 +
\frac{1}{2}
 +
R _ {jkl}  ^ {i} \omega  ^ {k} \wedge \omega  ^ {l} .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515065.png" /> form, respectively, the torsion and curvature tensors of the affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515066.png" />. The last two equations for the components of the connection form are called the structure equations for the affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025150/c02515067.png" />.
+
Here $  T _ {jk} ^ { i } $
 +
and $  R _ {jkl}  ^ {i} $
 +
form, respectively, the torsion and curvature tensors of the affine connection on $  M $.  
 +
The last two equations for the components of the connection form are called the structure equations for the affine connection on $  M $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cartan,  "Espaces à connexion affine, projective et conforme"  ''Acta Math.'' , '''48'''  (1926)  pp. 1–42</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.F. Laptev,  "Differential geometry of imbedded manifolds. Group-theoretical method of differential-geometric investigations"  ''Trudy Moskov. Mat. Obshch.'' , '''2'''  (1953)  pp. 275–382  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''2''' , Interscience  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cartan,  "Espaces à connexion affine, projective et conforme"  ''Acta Math.'' , '''48'''  (1926)  pp. 1–42</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.F. Laptev,  "Differential geometry of imbedded manifolds. Group-theoretical method of differential-geometric investigations"  ''Trudy Moskov. Mat. Obshch.'' , '''2'''  (1953)  pp. 275–382  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''2''' , Interscience  (1969)</TD></TR></table>

Latest revision as of 17:46, 4 June 2020


A linear differential form $ \theta $ on a principal fibre bundle $ P $ that takes values in the Lie algebra $ \mathfrak g $ of the structure group $ G $ of $ P $. It is defined by a certain linear connection $ \Gamma $ on $ P $, and it determines this connection uniquely. The values of the connection form $ \theta _ {y} ( Y) $ in terms of $ \Gamma $, where $ y \in P $ and $ Y \in T _ {y} ( P) $, are defined as the elements of $ \mathfrak g $ which, under the action of $ G $ on $ P $, generate the second component of $ Y $ relative to the direct sum $ T _ {y} ( F) = \Delta _ {y} \oplus T _ {y} ( G _ {y} ) $. Here $ G _ {y} $ is the fibre of $ P $ that contains $ y $ and $ \Delta $ is the horizontal distribution of $ \Gamma $. The horizontal distribution $ \Delta $, and so the connection $ \Gamma $, can be recovered from the connection form $ \theta $ in the following way.

The Cartan–Laptev theorem. For a form $ \theta $ on $ P $ with values in $ \mathfrak g $ to be a connection form it is necessary and sufficient that: 1) $ \theta _ {y} ( Y) $, for $ Y \in T _ {y} ( G _ {y} ) $, is the element of $ \mathfrak g $ that generates $ Y $ under the action of $ G $ on $ P $; and 2) the $ \mathfrak g $- valued $ 2 $- form

$$ \Omega = d \theta + \frac{1}{2} [ \theta , \theta ] , $$

formed from $ \theta $, is semi-basic, or horizontal, that is, $ \Omega _ {y} ( Y , Y _ {1} ) = 0 $ if at least one of the vectors $ Y , Y _ {1} $ belongs to $ T _ {y} ( G _ {y} ) $. The $ 2 $- form $ \Omega $ is called the curvature form of the connection. If a basis $ \{ e _ {1} \dots e _ {r} \} $ is defined in $ \mathfrak g $, then condition 2) can locally be expressed by the equalities:

$$ d \theta ^ \rho + \frac{1}{2} C _ {\sigma \tau } ^ \rho \theta ^ \sigma \wedge \theta ^ \tau = \ \frac{1}{2} R _ {ij} ^ \rho \omega ^ {i} \wedge \omega ^ {j} , $$

where $ \omega ^ {1} \dots \omega ^ {n} $ are certain linearly independent semi-basic $ 1 $- forms. The necessity of condition 2) was established in this form by E. Cartan [1]; its sufficiency under the additional assumption of 1) was proved by G.F. Laptev [2]. The equations

for the components of the connection form are called the structure equations for the connection in $ P $, the $ R _ {ij} ^ \rho $ define the curvature object.

As an example, let $ P $ be the space of affine frames in the tangent bundle of an $ n $- dimensional smooth manifold $ M $. Then $ G $ and $ \mathfrak g $ are, respectively, the group and the Lie algebra of matrices of the form

$$ \left \| \begin{array}{cc} 1 &a ^ {i} \\ 0 &A _ {j} ^ {i} \\ \end{array} \right \| ,\ \ \mathop{\rm det} | A _ {j} ^ {i} | \neq 0 , $$

and

$$ \left \| \begin{array}{cc} 0 &\mathfrak g ^ {i} \\ 0 &\mathfrak g _ {j} ^ {i} \\ \end{array} \right \| \ \ ( i , j = 1 \dots n ) . $$

By the Cartan–Laptev theorem, the $ \mathfrak g $- valued $ 1 $- form

$$ \theta = \ \left \| \begin{array}{cc} 0 &\omega ^ {i} \\ 0 &\omega _ {j} ^ {i} \\ \end{array} \right \| $$

on $ P $ is the connection form of a certain affine connection on $ M $ if and only if

$$ d \omega ^ {i} + \omega _ {j} ^ {i} \wedge \omega ^ {j} = \ \frac{1}{2} T _ {jk} ^ { i } \omega ^ {j} \wedge \omega ^ {k} , $$

$$ d \omega _ {j} ^ {i} = \omega _ {k} ^ {i} \wedge \omega _ {j} ^ {k} = \frac{1}{2} R _ {jkl} ^ {i} \omega ^ {k} \wedge \omega ^ {l} . $$

Here $ T _ {jk} ^ { i } $ and $ R _ {jkl} ^ {i} $ form, respectively, the torsion and curvature tensors of the affine connection on $ M $. The last two equations for the components of the connection form are called the structure equations for the affine connection on $ M $.

References

[1] E. Cartan, "Espaces à connexion affine, projective et conforme" Acta Math. , 48 (1926) pp. 1–42
[2] G.F. Laptev, "Differential geometry of imbedded manifolds. Group-theoretical method of differential-geometric investigations" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 275–382 (In Russian)
[3] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969)
How to Cite This Entry:
Connection form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connection_form&oldid=13808
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article