Namespaces
Variants
Actions

Difference between revisions of "Affine connection"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex done)
 
Line 1: Line 1:
A [[Differential-geometric structure|differential-geometric structure]] on a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a0109501.png" />, a special kind of connection on a manifold (cf. [[Connections on a manifold|Connections on a manifold]]), when the smooth fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a0109502.png" /> attached to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a0109503.png" /> has the affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a0109504.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a0109505.png" /> as its typical fibre. The structure of such an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a0109506.png" /> involves the assignment to each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a0109507.png" /> of a copy of the affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a0109508.png" />, which is identified with the tangent centro-affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a0109509.png" />. In an affine connection each smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095010.png" /> with origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095011.png" /> and each one of its points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095012.png" /> is thus provided with an affine mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095013.png" /> which satisfies the condition formulated below. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095014.png" /> be covered with coordinate domains, each provided with a smooth field of affine frames in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095015.png" />. The origin of these frames coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095016.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095017.png" /> smooth vector fields, linearly independent at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095018.png" /> of the domain, are given). The requirement is that, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095019.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095020.png" /> moves along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095021.png" /> towards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095022.png" />, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095023.png" /> tends to become the identity mapping, and that the principal part of its deviation from the identity mapping be defined, with respect to some frame, by the system of linear differential forms
+
{{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/a/a010/a010950/a01095024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
A [[Differential-geometric structure|differential-geometric structure]] on a smooth manifold  $  M $,
 +
a special kind of connection on a manifold (cf. [[Connections on a manifold|Connections on a manifold]]), when the smooth fibre bundle  $  E $
 +
attached to  $  M $
 +
has the affine space  $  A _ {n} $
 +
of dimension  $  n = { \mathop{\rm dim}\nolimits} \  M $
 +
as its typical fibre. The structure of such an  $  E $
 +
involves the assignment to each point  $  x \in M $
 +
of a copy of the affine space  $  ( A _ {n} ) _ {x} $,
 +
which is identified with the tangent centro-affine space  $  T _ {x} (M) $.
 +
In an affine connection each smooth curve  $  L \in M $
 +
with origin  $  x _ {0} $
 +
and each one of its points  $  x _ {t} $
 +
is thus provided with an affine mapping  $  ( A _ {n} ) _ {x _ t}  \rightarrow ( A _ {n} ) _ {x _ 0}  $
 +
which satisfies the condition formulated below. Let  $  M $
 +
be covered with coordinate domains, each provided with a smooth field of affine frames in  $  (A _ {n} ) _ {x} $.
 +
The origin of these frames coincides with  $  x $(
 +
i.e. $  n $
 +
smooth vector fields, linearly independent at each point  $  x $
 +
of the domain, are given). The requirement is that, as  $  t \rightarrow 0 $,
 +
when  $  x _ {t} $
 +
moves along  $  L $
 +
towards  $  x _ {0} $,
 +
the mapping  $  (A _ {n} ) _ {x _ t}  \rightarrow (A _ {n} ) _ { x _ 0 } $
 +
tends to become the identity mapping, and that the principal part of its deviation from the identity mapping be defined, with respect to some frame, by the system of linear differential forms
  
Thus, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095025.png" />, the image of the frame at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095026.png" /> is the system consisting of the point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095027.png" /> with position vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095029.png" /> vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095031.png" /> is the tangent vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095032.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095033.png" />, and
+
$$ \tag{1}
 +
\left .
 +
{
 +
{\omega  ^ {i} \  = \  \Gamma _ {k}  ^ {i} \  dx  ^ {k} ,\ \
 +
\mathop{\rm det}\nolimits \  | \Gamma _ {k}  ^ {i} | \  \neq \  0,} \atop {\omega _ {j}  ^ {i} \  = \  \Gamma _ {jk}  ^ {i} \omega  ^ {k} .}}
 +
\right \}
 +
$$
  
<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/a/a010/a010950/a01095034.png" /></td> </tr></table>
+
Thus, for  $  ( A _ {n} ) _ {x _ t}  \rightarrow ( A _ {n} ) _ {x _ 0}  $,
 +
the image of the frame at  $  x _ {t} $
 +
is the system consisting of the point in  $  (A _ {n} ) _ {x _ 0}  $
 +
with position vector  $  e _ {i} [ \omega  ^ {i} (X) t + \epsilon  ^ {i} (t) ] $
 +
and  $  n $
 +
vectors  $  e _ {i} [ \delta _ {j}  ^ {i} + \omega _ {j}  ^ {i} (X) t + \epsilon _ {j}  ^ {i} (t) ] $,
 +
where  $  X $
 +
is the tangent vector to  $  L $
 +
at  $  x _ {0} $,
 +
and
  
A manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095035.png" /> with an affine connection defined on it is called a space with an affine connection. During the transformation of a frame of the field at an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095036.png" /> according to the formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095038.png" />, i.e. when passing to an arbitrary element of the principal fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095039.png" /> of frames in the tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095040.png" /> with origins at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095041.png" />, the forms (1) are replaced by the following <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095042.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095043.png" />:
+
$$
 +
\lim\limits _ {t \rightarrow 0} \ 
 +
\frac{\epsilon  ^ {i} (t)}{t}
 +
= 0 ,\ \
 +
\lim\limits _ {t \rightarrow 0} \ 
 +
\frac{\epsilon _ {j}  ^ {i} (t)}{t}
 +
= 0.
 +
$$
  
<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/a/a010/a010950/a01095044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
A manifold  $  M $
 +
with an affine connection defined on it is called a space with an affine connection. During the transformation of a frame of the field at an arbitrary point  $  x \in M $
 +
according to the formulas  $  e _ {i ^ \prime}  = A _ {i ^ \prime}  ^ {j} e _ {j} $,
 +
$  e _ {j} = A _ {j} ^ {i ^ \prime} e _ {i ^ \prime}  $,
 +
i.e. when passing to an arbitrary element of the principal fibre bundle  $  P $
 +
of frames in the tangent spaces  $  ( A _ {n} ) _ {x} $
 +
with origins at the point  $  x $,
 +
the forms (1) are replaced by the following  $  1 $-
 +
forms on  $  P $:
  
while the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095045.png" />-forms
+
$$ \tag{2}
 +
\left .
 +
{
 +
{\omega ^ {i ^ \prime} \  = \  A _ {j} ^ {i ^ \prime} \omega  ^ {j} ,} \atop {\omega _ {j ^ \prime} ^ {i ^ \prime} \  = \
 +
A _ {k} ^ {i ^ \prime} \  d A _ {j ^ \prime}  ^ {k} +
 +
A _ {k} ^ {i ^ \prime} A _ {j ^ \prime}  ^ {l} \omega
 +
_ {l}  ^ {k} ,}
 +
}
 +
\right \}
 +
$$
  
<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/a/a010/a010950/a01095046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
while the  $  2 $-
 +
forms
 +
 
 +
$$ \tag{3}
 +
\left .
 +
{
 +
{\Omega  ^ {i} \  = \  d \omega  ^ {i} + \omega _ {j}  ^ {i} \wedge
 +
\omega  ^ {j} ,} \atop {\Omega _ {j}  ^ {i} \  = \  d \omega _ {j}  ^ {i} + \omega  _ {k}  ^ {i} \wedge \omega _ {j}  ^ {k} ,}}
 +
\right \}
 +
$$
  
 
are transformed as follows:
 
are transformed as follows:
  
<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/a/a010/a010950/a01095047.png" /></td> </tr></table>
+
$$
 +
\Omega ^ {i ^ \prime} \  = \  A _ {j} ^ {i ^ \prime} \Omega  ^ {j} ,\ \
 +
\Omega _ {j ^ \prime}  ^ {i ^ \prime} \  = \  A _ {k} ^ {i ^ \prime}
 +
A _ {j ^ \prime}  ^ {l} \Omega _ {l}  ^ {k} ,
 +
$$
 +
 
 +
where  $  \Omega ^ {i ^ \prime} $
 +
and  $  \Omega _ {j ^ \prime}  ^ {i ^ \prime} $
 +
are composed from the forms (2) according to (3). The equations (3) are called the structure equations of the affine connection on  $  M $.
 +
Here the left-hand sides — the so-called torsion forms  $  \Omega  ^ {i} $
 +
and curvature forms  $  \Omega _ {j}  ^ {i} $
 +
— are semi-basic (cf. [[Torsion form|Torsion form]]; [[Curvature form|Curvature form]]), i.e. they are linear combinations of the  $  \omega  ^ {k} \wedge \omega  ^ {l} $:
 +
 
 +
$$ \tag{4}
 +
\left .
 +
{
 +
{\Omega  ^ {i} \  =
 +
\frac{1}{2}
 +
S _ {jk}  ^ {i} \omega  ^ {j} \wedge
 +
\omega  ^ {k} ,} \atop {\Omega _ {j}  ^ {i} \  =
 +
\frac{1}{2}
 +
R _ {jkl}  ^ {i} \omega  ^ {k} \wedge \omega  ^ {l} .}}
 +
\right \}
 +
$$
 +
 
 +
All  $  1 $-
 +
forms  $  \omega  ^ {i} $
 +
and  $  \omega _ {j}  ^ {i} $,
 +
defined on  $  P $
 +
and satisfying equations (3) with left-hand sides of type (4), define a certain affine connection on  $  M $.
 +
The mapping  $  ( A _ {n} ) _ {x _ t}  \rightarrow ( A _ {n} ) _ {x _ 0}  $
 +
for a curve  $  L \in M $
 +
is obtained as follows. A smooth field of frames is chosen in a coordinate neighbourhood of the origin  $  x _ {0} $
 +
of the curve  $  L $,
 +
and the image of the frame at point  $  x _ {t} $
 +
is defined as the solution  $  \{ x(t),\  e _ {i} (t) \} $
 +
of the system
 +
 
 +
$$ \tag{5}
 +
\left .
 +
{
 +
{du \  = \  ( \omega  ^ {i} )  _ {x(t)} ( \dot{x} (t) ) u _ {i} ,} \atop {du _ {j} \  = \  ( \omega _ {j}  ^ {i} )  _ {x(t)} ( \dot{x} (t))
 +
u _ {i} ,}}
 +
\right \}
 +
$$
 +
 
 +
for the initial conditions  $  u(0) = 0,\  u _ {i} (0) = e _ {i} $,
 +
where  $  x  ^ {i} = x  ^ {i} (t) $
 +
are the defining equations of the curve  $  L $.
 +
The curve which is described in  $  ( A _ {n} ) _ {x _ 0}  $
 +
by the point with position vector  $  x(t) $
 +
with respect to  $  x _ {0} $
 +
is known as the development of  $  L $.  
 +
The field of frames in the coordinate neighbourhood may be so chosen that  $  \omega  ^ {i} = d x  ^ {i} $;
 +
then  $  \omega _ {j}  ^ {i} = \Gamma _ {jk}  ^ {i} \  d x  ^ {k} $.  
 +
In the intersection of the coordinate neighbourhoods,  $  dx ^ {i ^ \prime} = ( {\partial  x ^ {i ^ \prime}} / \partial  x  ^ {j} ) \omega  ^ {j} $,
 +
i.e. $  A _ {j} ^ {i ^ \prime} = {\partial  x ^ {i ^ \prime}} / \partial  x  ^ {j} $
 +
and
 +
 
 +
$$ \tag{6}
 +
\Gamma _ {j  ^  \prime  k ^ \prime} ^ {i ^ \prime} \  = \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095049.png" /> are composed from the forms (2) according to (3). The equations (3) are called the structure equations of the affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095050.png" />. Here the left-hand sides — the so-called torsion forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095051.png" /> and curvature forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095052.png" /> — are semi-basic (cf. [[Torsion form|Torsion form]]; [[Curvature form|Curvature form]]), i.e. they are linear combinations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095053.png" />:
+
\frac{\partial  x ^ {i ^ \prime}}{\partial  x ^ 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/a/a010/a010950/a01095054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\frac{\partial  x ^ j}{\partial  x ^ {j ^ \prime}}
  
All <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095055.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095057.png" />, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095058.png" /> and satisfying equations (3) with left-hand sides of type (4), define a certain affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095059.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095060.png" /> for a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095061.png" /> is obtained as follows. A smooth field of frames is chosen in a coordinate neighbourhood of the origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095062.png" /> of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095063.png" />, and the image of the frame at point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095064.png" /> is defined as the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095065.png" /> of the system
+
\frac{\partial  x ^ k}{\partial  x ^ {k ^ \prime}}
  
<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/a/a010/a010950/a01095066.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
\Gamma _ {jk}  ^ {i} +
 +
\frac{\partial  ^ {2} x ^ i}{\partial  x ^ {j ^ \prime} \partial  x ^ {k ^ \prime}}
  
for the initial conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095068.png" /> are the defining equations of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095069.png" />. The curve which is described in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095070.png" /> by the point with position vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095071.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095072.png" /> is known as the development of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095073.png" />. The field of frames in the coordinate neighbourhood may be so chosen that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095074.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095075.png" />. In the intersection of the coordinate neighbourhoods, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095076.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095077.png" /> and
+
\frac{\partial  x ^ {i ^ \prime}}{\partial  x ^ 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/a/a010/a010950/a01095078.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{7}
 +
\left . { {S _ {jk}  ^ {i} \  = \  \Gamma _ {jk}  ^ {i} - \Gamma
 +
_ {kj}  ^ {i} ,} \atop {R _ {j k l}  ^ {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/a/a010/a010950/a01095079.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
\frac{\partial  \Gamma _ {jl} ^ i}{\partial  x ^ k}
 +
-
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095081.png" /> are, respectively, the [[Torsion tensor|torsion tensor]] and the [[Curvature tensor|curvature tensor]] of the affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095082.png" />. An affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095083.png" /> may be defined by a system of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095084.png" /> on each coordinate neighbourhood which transforms in the intersection of two neighbourhoods according to formula (5). The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095085.png" /> is called the object of the affine connection. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095086.png" /> is obtained with the aid of (5) into which
+
\frac{\partial  \Gamma _ {jk} ^ i}{\partial  x ^ l}
 +
+
 +
\Gamma _ {pk}  ^ {i} \Gamma _ {jl}  ^ {p} - \Gamma  _ {pl}  ^ {i} \Gamma _ {jk}  ^ {p} .}} \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/a/a010/a010950/a01095087.png" /></td> </tr></table>
+
Here  $  S _ {jk}  ^ {i} $
 +
and  $  R _ {jkl}  ^ {i} $
 +
are, respectively, the [[Torsion tensor|torsion tensor]] and the [[Curvature tensor|curvature tensor]] of the affine connection on  $  M $.  
 +
An affine connection on  $  M $
 +
may be defined by a system of functions  $  \Gamma _ {jk}  ^ {i} $
 +
on each coordinate neighbourhood which transforms in the intersection of two neighbourhoods according to formula (5). The system  $  \Gamma _ {jk}  ^ {i} $
 +
is called the object of the affine connection. The mapping  $  ( A _ {n} ) _ {x _ t}  $
 +
is obtained with the aid of (5) into which
 +
 
 +
$$
 +
\omega  ^ {i} \  = \  dx  ^ {i} ,\ \
 +
\omega _ {j}  ^ {i} \  = \  \Gamma _ {jk}  ^ {i} \  dx  ^ {k} ,
 +
$$
  
 
is to be substituted.
 
is to be substituted.
  
If, in some neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095088.png" />, a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095089.png" /> is given, then, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095090.png" />, the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095091.png" /> is mapped into the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095092.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095093.png" /> is the solution of system (5)). The differential of this in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095094.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095095.png" />:
+
If, in some neighbourhood of the point $  x _ {0} $,  
 +
a vector field $  X = \xi  ^ {i} e _ {i} $
 +
is given, then, when $  ( A _ {n} ) _ {x _ t}  \rightarrow ( A _ {n} ) _ {x _ 0}  $,  
 +
the vector $  X _ {x(t)}  $
 +
is mapped into the vector $  \xi  ^ {i} ( x  ^ {t} ) e _ {i} (t) $(
 +
where $  \{ e _ {i} (t) \} $
 +
is the solution of system (5)). The differential of this in $  ( A _ {n} ) _ {x _ 0}  $
 +
at $  t = 0 $:
 +
 
 +
$$
 +
( d \xi  ^ {i} + \xi  ^ {i} \omega _ {j}  ^ {i} ) e _ {i} \  = \
 +
\left (
 +
\frac{\partial  \xi ^ i}{\partial  x ^ k}
 +
+ \xi  ^ {j} \Gamma  _ {jk}  ^ {i} \right ) \  dx  ^ {k} e _ {i} ,
 +
$$
  
<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/a/a010/a010950/a01095096.png" /></td> </tr></table>
+
is called the [[Covariant differential|covariant differential]] of the field  $  X $
 +
with respect to the given affine connection. Here
  
is called the [[Covariant differential|covariant differential]] of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095097.png" /> with respect to the given affine connection. Here
+
$$
 +
\nabla _ {k} \xi  ^ {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/a/a010/a010950/a01095098.png" /></td> </tr></table>
+
\frac{\partial  \xi ^ i}{\partial  x ^ k}
 +
+
 +
\xi  ^ {j} \Gamma _ {jk}  ^ {i}  $$
  
form a tensor field, called the [[Covariant derivative|covariant derivative]] of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a01095099.png" />. If a second vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950100.png" /> is given, the covariant derivative of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950101.png" /> in the direction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950102.png" /> is defined as
+
form a tensor field, called the [[Covariant derivative|covariant derivative]] of the field $  X = \xi  ^ {i} e _ {i} $.  
 +
If a second vector field $  Y = \eta  ^ {k} e _ {k} $
 +
is given, the covariant derivative of the field $  X $
 +
in the direction of $  Y $
 +
is defined as
  
<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/a/a010/a010950/a010950103.png" /></td> </tr></table>
+
$$
 +
\nabla _ {Y} X \  = \  \eta  ^ {k} \nabla _ {k} \xi  ^ {i} e _ {i} ,
 +
$$
  
 
which may also be defined with respect to an arbitrary field of frames by the formula
 
which may also be defined with respect to an arbitrary field of frames 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/a/a010/a010950/a010950104.png" /></td> </tr></table>
+
$$
 +
\omega  ^ {i} ( \nabla _ {Y} X ) \  = \  Y \omega  ^ {i} (X) +
 +
\omega _ {k}  ^ {i} (Y) \omega  ^ {k} (X).
 +
$$
  
An affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950105.png" /> may also be defined as a bilinear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950106.png" /> which assigns a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950107.png" /> to each two vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950109.png" />, and which possesses the properties:
+
An affine connection on $  M $
 +
may also be defined as a bilinear operator $  \nabla $
 +
which assigns a vector field $  \nabla _ {Y} X $
 +
to each two vector fields $  X $
 +
and $  Y $,  
 +
and which possesses the properties:
  
<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/a/a010/a010950/a010950110.png" /></td> </tr></table>
+
$$
 +
\nabla _ {Y} ( f X ) \  = \  ( Y f ) X + f \nabla _ {Y} X ,
 +
\ \  \nabla _ {fY} X \  = \  f \nabla _ {Y} X ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950111.png" /> is a smooth function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950112.png" />. The relation between these definitions is established by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950113.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950114.png" /> is the field of frames. The fields of the torsion tensor and curvature tensor
+
where $  f $
 +
is a smooth function on $  M $.  
 +
The relation between these definitions is established by the formula $  \nabla _ {e _ k}  e _ {j} = \Gamma _ {jk}  ^ {i} e _ {i} $
 +
where $  \{ e _ {i} \} $
 +
is the field of frames. The fields of the torsion tensor and curvature tensor
  
<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/a/a010/a010950/a010950115.png" /></td> </tr></table>
+
$$
 +
S ( X ,\  Y ) \  = \  S _ {jk}  ^ {i} \xi  ^ {j} \eta  ^ {k} e _ {i} \  = \
 +
\Omega  ^ {i} ( X ,\  Y ) e _ {i} ,
 +
$$
  
<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/a/a010/a010950/a010950116.png" /></td> </tr></table>
+
$$
 +
R ( X ,\  Y ) Z \  = \  ( R _ {jkl}  ^ {i} \xi  ^ {k} \eta  ^ {l} )
 +
\xi  ^ {j} e _ {i} \  = \  \Omega _ {j}  ^ {i} ( X ,\  Y ) \omega  ^ {j} ( Z ) e  _ {i}  $$
  
 
are defined by the formulas:
 
are defined 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/a/a010/a010950/a010950117.png" /></td> </tr></table>
+
$$
 +
S ( X ,\  Y ) \  = \  \nabla _ {X} Y - \nabla _ {Y} X - [ X ,\  Y ],
 +
$$
 +
 
 +
$$
 +
R ( X ,\  Y ) \  = \  \nabla _ {X} \nabla _ {Y} Z - \nabla  _ {Y} \nabla _ {X} Z - \nabla _ {[ X ,\  Y ]}  Z .
 +
$$
  
<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/a/a010/a010950/a010950118.png" /></td> </tr></table>
+
A vector field  $  X $
 +
is said to be parallel along the curve  $  L $
 +
if  $  \nabla _ {\dot{x} (t)}  X _ {x(t)}  = 0 $
 +
holds identically with respect to  $  t $,
 +
i.e. if, along  $  L $,
  
A vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950119.png" /> is said to be parallel along the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950120.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950121.png" /> holds identically with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950122.png" />, i.e. if, along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950123.png" />,
+
$$
 +
d \xi  ^ {i} + \xi  ^ {j} \omega _ {j}  ^ {i} \  = 0 .
 +
$$
  
<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/a/a010/a010950/a010950124.png" /></td> </tr></table>
+
Parallel vector fields are used to effect [[Parallel displacement(2)|parallel displacement]] of vectors (and, generally, of tensors) in an affine connection, representing a linear mapping of the tangent vector spaces  $  T _ {x _ t}  (M) \rightarrow T _ {x _ 0}  (M) $,
 +
defined by the mapping  $  ( A _ {n} ) _ {x _ t}  \rightarrow ( A _ {n} ) _ {x _ 0}  $.  
 +
In this sense any affine connection generates a [[Linear connection|linear connection]] on  $  M $.
  
Parallel vector fields are used to effect [[Parallel displacement(2)|parallel displacement]] of vectors (and, generally, of tensors) in an affine connection, representing a linear mapping of the tangent vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950125.png" />, defined by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950126.png" />. In this sense any affine connection generates a [[Linear connection|linear connection]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950127.png" />.
+
A curve  $  L $
 +
is called a geodesic line in a given affine connection if its development is a straight line; in other words, if, by a suitable parametrization, its tangent vector field  $  \dot{x} (t) $
 +
is parallel to it. Geodesic lines are defined with respect to a local coordinate system by the system
  
A curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950128.png" /> is called a geodesic line in a given affine connection if its development is a straight line; in other words, if, by a suitable parametrization, its tangent vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950129.png" /> is parallel to it. Geodesic lines are defined with respect to a local coordinate system by the system
+
$$
  
<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/a/a010/a010950/a010950130.png" /></td> </tr></table>
+
\frac{d  ^ {2} x ^ i}{d t ^ 2}
 +
+ \Gamma _ {jk}  ^ {i}
 +
\frac{d x ^ j}{dt}
 +
 +
\frac{d x ^ k}{dt}
 +
= 0.
 +
$$
  
 
Through each point, in each direction passes one geodesic line.
 
Through each point, in each direction passes one geodesic line.
  
There is a one-to-one correspondence between affine connections on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950131.png" /> and connections in principal fibre bundles of free affine frames in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950132.png" />, generated by them. To closed curves with origin and end at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950133.png" /> there correspond affine transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950134.png" />, which form the non-homogeneous [[Holonomy group|holonomy group]] of the given affine connection. The corresponding linear automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950135.png" /> form the homogeneous holonomy group. In accordance with the holonomy theorem, the Lie algebras of these groups are defined by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950136.png" />-forms of torsion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950137.png" /> and curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950138.png" />. The Bianchi identities apply to the latter:
+
There is a one-to-one correspondence between affine connections on $  M $
 +
and connections in principal fibre bundles of free affine frames in $  (A _ {n} ) _ {x} ,\  x \in M $,  
 +
generated by them. To closed curves with origin and end at $  x $
 +
there correspond affine transformations $  ( A _ {n} ) _ {x} \rightarrow ( A _ {n} ) _ {x} $,  
 +
which form the non-homogeneous [[Holonomy group|holonomy group]] of the given affine connection. The corresponding linear automorphisms $  T _ {x} (M) \rightarrow T _ {x} (M) $
 +
form the homogeneous holonomy group. In accordance with the holonomy theorem, the Lie algebras of these groups are defined by the $  2 $-
 +
forms of torsion $  \Omega  ^ {i} $
 +
and curvature $  \Omega _ {j}  ^ {i} $.  
 +
The Bianchi identities apply to the latter:
  
<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/a/a010/a010950/a010950139.png" /></td> </tr></table>
+
$$
 +
d \Omega  ^ {i} \  = \  \Omega _ {j}  ^ {i} \wedge \omega  ^ {i} - \omega _ {j}  ^ {i} \wedge \Omega  ^ {j} ,\ \
 +
d \Omega _ {j}  ^ {i} \  = \  \Omega _ {k}  ^ {i} \wedge \omega _ {j}  ^ {k} - \omega _ {k}  ^ {i} \wedge \Omega _ {j}  ^ {k} .
 +
$$
  
In particular, for torsion-free affine connections, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010950/a010950140.png" />, these identities reduce to the following:
+
In particular, for torsion-free affine connections, when $  \Omega  ^ {i} = 0 $,  
 +
these identities reduce to the following:
  
<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/a/a010/a010950/a010950141.png" /></td> </tr></table>
+
$$
 +
R _ {jkl}  ^ {i} + R _ {klj}  ^ {i} + R _ {ljk}  ^ {i} \  = \  0 ,\ \
 +
\nabla _ {m} R _ {jkl}  ^ {i} + \nabla _ {k} R _ {jlm}  ^ {i} +
 +
\nabla _ {l} R _ {jmk}  ^ {i} \  = 0 .
 +
$$
  
 
The concept of an affine connection arose in 1917 in Riemannian geometry (in the form of the [[Levi-Civita connection|Levi-Civita connection]]); it found an independent meaning in 1918–1924 owing to work of H. Weyl [[#References|[1]]] and E. Cartan .
 
The concept of an affine connection arose in 1917 in Riemannian geometry (in the form of the [[Levi-Civita connection|Levi-Civita connection]]); it found an independent meaning in 1918–1924 owing to work of H. Weyl [[#References|[1]]] and E. Cartan .
Line 95: Line 329:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Weyl,  "Raum, Zeit, Materie" , Springer  (1923)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  E. Cartan,  "Sur les variétés a connexion affine et la théorie de la relativité généralisée (première partie)"  ''Ann. Sci. École Norm. Sup.'' , '''40'''  (1923)  pp. 325–412</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  E. Cartan,  "Sur les variétés a connexion affine et la théorie de la relativité généralisée (première partie suite)"  ''Ann. Sci. École Norm. Sup.'' , '''41'''  (1924)  pp. 1–25</TD></TR><TR><TD valign="top">[2c]</TD> <TD valign="top">  E. Cartan,  "Sur les variétés a connexion affine et la théorie de la relativité généralisée (deuxième partie)"  ''Ann. Sci. École Norm. Sup.'' , '''42'''  (1925)  pp. 17–88</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  E. Cartan,  "Sur les variétés à connexion projective"  ''Bull. Soc. Math. France'' , '''52'''  (1924)  pp. 205–241</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  E. Cartan,  "Sur les espaces à connexion conforme"  ''Ann. Soc. Polon. Math.'' , '''2'''  (1923)  pp. 171–221</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.M. Postnikov,  "The variational theory of geodesics" , Saunders  (1967)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Weyl,  "Raum, Zeit, Materie" , Springer  (1923)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  E. Cartan,  "Sur les variétés a connexion affine et la théorie de la relativité généralisée (première partie)"  ''Ann. Sci. École Norm. Sup.'' , '''40'''  (1923)  pp. 325–412</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  E. Cartan,  "Sur les variétés a connexion affine et la théorie de la relativité généralisée (première partie suite)"  ''Ann. Sci. École Norm. Sup.'' , '''41'''  (1924)  pp. 1–25</TD></TR><TR><TD valign="top">[2c]</TD> <TD valign="top">  E. Cartan,  "Sur les variétés a connexion affine et la théorie de la relativité généralisée (deuxième partie)"  ''Ann. Sci. École Norm. Sup.'' , '''42'''  (1925)  pp. 17–88</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  E. Cartan,  "Sur les variétés à connexion projective"  ''Bull. Soc. Math. France'' , '''52'''  (1924)  pp. 205–241</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  E. Cartan,  "Sur les espaces à connexion conforme"  ''Ann. Soc. Polon. Math.'' , '''2'''  (1923)  pp. 171–221</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.M. Postnikov,  "The variational theory of geodesics" , Saunders  (1967)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 11:10, 6 February 2020


A differential-geometric structure on a smooth manifold $ M $, a special kind of connection on a manifold (cf. Connections on a manifold), when the smooth fibre bundle $ E $ attached to $ M $ has the affine space $ A _ {n} $ of dimension $ n = { \mathop{\rm dim}\nolimits} \ M $ as its typical fibre. The structure of such an $ E $ involves the assignment to each point $ x \in M $ of a copy of the affine space $ ( A _ {n} ) _ {x} $, which is identified with the tangent centro-affine space $ T _ {x} (M) $. In an affine connection each smooth curve $ L \in M $ with origin $ x _ {0} $ and each one of its points $ x _ {t} $ is thus provided with an affine mapping $ ( A _ {n} ) _ {x _ t} \rightarrow ( A _ {n} ) _ {x _ 0} $ which satisfies the condition formulated below. Let $ M $ be covered with coordinate domains, each provided with a smooth field of affine frames in $ (A _ {n} ) _ {x} $. The origin of these frames coincides with $ x $( i.e. $ n $ smooth vector fields, linearly independent at each point $ x $ of the domain, are given). The requirement is that, as $ t \rightarrow 0 $, when $ x _ {t} $ moves along $ L $ towards $ x _ {0} $, the mapping $ (A _ {n} ) _ {x _ t} \rightarrow (A _ {n} ) _ { x _ 0 } $ tends to become the identity mapping, and that the principal part of its deviation from the identity mapping be defined, with respect to some frame, by the system of linear differential forms

$$ \tag{1} \left . { {\omega ^ {i} \ = \ \Gamma _ {k} ^ {i} \ dx ^ {k} ,\ \ \mathop{\rm det}\nolimits \ | \Gamma _ {k} ^ {i} | \ \neq \ 0,} \atop {\omega _ {j} ^ {i} \ = \ \Gamma _ {jk} ^ {i} \omega ^ {k} .}} \right \} $$

Thus, for $ ( A _ {n} ) _ {x _ t} \rightarrow ( A _ {n} ) _ {x _ 0} $, the image of the frame at $ x _ {t} $ is the system consisting of the point in $ (A _ {n} ) _ {x _ 0} $ with position vector $ e _ {i} [ \omega ^ {i} (X) t + \epsilon ^ {i} (t) ] $ and $ n $ vectors $ e _ {i} [ \delta _ {j} ^ {i} + \omega _ {j} ^ {i} (X) t + \epsilon _ {j} ^ {i} (t) ] $, where $ X $ is the tangent vector to $ L $ at $ x _ {0} $, and

$$ \lim\limits _ {t \rightarrow 0} \ \frac{\epsilon ^ {i} (t)}{t} \ = \ 0 ,\ \ \lim\limits _ {t \rightarrow 0} \ \frac{\epsilon _ {j} ^ {i} (t)}{t} \ = \ 0. $$

A manifold $ M $ with an affine connection defined on it is called a space with an affine connection. During the transformation of a frame of the field at an arbitrary point $ x \in M $ according to the formulas $ e _ {i ^ \prime} = A _ {i ^ \prime} ^ {j} e _ {j} $, $ e _ {j} = A _ {j} ^ {i ^ \prime} e _ {i ^ \prime} $, i.e. when passing to an arbitrary element of the principal fibre bundle $ P $ of frames in the tangent spaces $ ( A _ {n} ) _ {x} $ with origins at the point $ x $, the forms (1) are replaced by the following $ 1 $- forms on $ P $:

$$ \tag{2} \left . { {\omega ^ {i ^ \prime} \ = \ A _ {j} ^ {i ^ \prime} \omega ^ {j} ,} \atop {\omega _ {j ^ \prime} ^ {i ^ \prime} \ = \ A _ {k} ^ {i ^ \prime} \ d A _ {j ^ \prime} ^ {k} + A _ {k} ^ {i ^ \prime} A _ {j ^ \prime} ^ {l} \omega _ {l} ^ {k} ,} } \right \} $$

while the $ 2 $- forms

$$ \tag{3} \left . { {\Omega ^ {i} \ = \ d \omega ^ {i} + \omega _ {j} ^ {i} \wedge \omega ^ {j} ,} \atop {\Omega _ {j} ^ {i} \ = \ d \omega _ {j} ^ {i} + \omega _ {k} ^ {i} \wedge \omega _ {j} ^ {k} ,}} \right \} $$

are transformed as follows:

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

where $ \Omega ^ {i ^ \prime} $ and $ \Omega _ {j ^ \prime} ^ {i ^ \prime} $ are composed from the forms (2) according to (3). The equations (3) are called the structure equations of the affine connection on $ M $. Here the left-hand sides — the so-called torsion forms $ \Omega ^ {i} $ and curvature forms $ \Omega _ {j} ^ {i} $ — are semi-basic (cf. Torsion form; Curvature form), i.e. they are linear combinations of the $ \omega ^ {k} \wedge \omega ^ {l} $:

$$ \tag{4} \left . { {\Omega ^ {i} \ = \ \frac{1}{2} S _ {jk} ^ {i} \omega ^ {j} \wedge \omega ^ {k} ,} \atop {\Omega _ {j} ^ {i} \ = \ \frac{1}{2} R _ {jkl} ^ {i} \omega ^ {k} \wedge \omega ^ {l} .}} \right \} $$

All $ 1 $- forms $ \omega ^ {i} $ and $ \omega _ {j} ^ {i} $, defined on $ P $ and satisfying equations (3) with left-hand sides of type (4), define a certain affine connection on $ M $. The mapping $ ( A _ {n} ) _ {x _ t} \rightarrow ( A _ {n} ) _ {x _ 0} $ for a curve $ L \in M $ is obtained as follows. A smooth field of frames is chosen in a coordinate neighbourhood of the origin $ x _ {0} $ of the curve $ L $, and the image of the frame at point $ x _ {t} $ is defined as the solution $ \{ x(t),\ e _ {i} (t) \} $ of the system

$$ \tag{5} \left . { {du \ = \ ( \omega ^ {i} ) _ {x(t)} ( \dot{x} (t) ) u _ {i} ,} \atop {du _ {j} \ = \ ( \omega _ {j} ^ {i} ) _ {x(t)} ( \dot{x} (t)) u _ {i} ,}} \right \} $$

for the initial conditions $ u(0) = 0,\ u _ {i} (0) = e _ {i} $, where $ x ^ {i} = x ^ {i} (t) $ are the defining equations of the curve $ L $. The curve which is described in $ ( A _ {n} ) _ {x _ 0} $ by the point with position vector $ x(t) $ with respect to $ x _ {0} $ is known as the development of $ L $. The field of frames in the coordinate neighbourhood may be so chosen that $ \omega ^ {i} = d x ^ {i} $; then $ \omega _ {j} ^ {i} = \Gamma _ {jk} ^ {i} \ d x ^ {k} $. In the intersection of the coordinate neighbourhoods, $ dx ^ {i ^ \prime} = ( {\partial x ^ {i ^ \prime}} / \partial x ^ {j} ) \omega ^ {j} $, i.e. $ A _ {j} ^ {i ^ \prime} = {\partial x ^ {i ^ \prime}} / \partial x ^ {j} $ and

$$ \tag{6} \Gamma _ {j ^ \prime k ^ \prime} ^ {i ^ \prime} \ = \ \frac{\partial x ^ {i ^ \prime}}{\partial x ^ i} \frac{\partial x ^ j}{\partial x ^ {j ^ \prime}} \frac{\partial x ^ k}{\partial x ^ {k ^ \prime}} \Gamma _ {jk} ^ {i} + \frac{\partial ^ {2} x ^ i}{\partial x ^ {j ^ \prime} \partial x ^ {k ^ \prime}} \frac{\partial x ^ {i ^ \prime}}{\partial x ^ i} , $$

$$ \tag{7} \left . { {S _ {jk} ^ {i} \ = \ \Gamma _ {jk} ^ {i} - \Gamma _ {kj} ^ {i} ,} \atop {R _ {j k l} ^ {i} \ = \ \frac{\partial \Gamma _ {jl} ^ i}{\partial x ^ k} - \frac{\partial \Gamma _ {jk} ^ i}{\partial x ^ l} + \Gamma _ {pk} ^ {i} \Gamma _ {jl} ^ {p} - \Gamma _ {pl} ^ {i} \Gamma _ {jk} ^ {p} .}} \right \} $$

Here $ S _ {jk} ^ {i} $ and $ R _ {jkl} ^ {i} $ are, respectively, the torsion tensor and the curvature tensor of the affine connection on $ M $. An affine connection on $ M $ may be defined by a system of functions $ \Gamma _ {jk} ^ {i} $ on each coordinate neighbourhood which transforms in the intersection of two neighbourhoods according to formula (5). The system $ \Gamma _ {jk} ^ {i} $ is called the object of the affine connection. The mapping $ ( A _ {n} ) _ {x _ t} $ is obtained with the aid of (5) into which

$$ \omega ^ {i} \ = \ dx ^ {i} ,\ \ \omega _ {j} ^ {i} \ = \ \Gamma _ {jk} ^ {i} \ dx ^ {k} , $$

is to be substituted.

If, in some neighbourhood of the point $ x _ {0} $, a vector field $ X = \xi ^ {i} e _ {i} $ is given, then, when $ ( A _ {n} ) _ {x _ t} \rightarrow ( A _ {n} ) _ {x _ 0} $, the vector $ X _ {x(t)} $ is mapped into the vector $ \xi ^ {i} ( x ^ {t} ) e _ {i} (t) $( where $ \{ e _ {i} (t) \} $ is the solution of system (5)). The differential of this in $ ( A _ {n} ) _ {x _ 0} $ at $ t = 0 $:

$$ ( d \xi ^ {i} + \xi ^ {i} \omega _ {j} ^ {i} ) e _ {i} \ = \ \left ( \frac{\partial \xi ^ i}{\partial x ^ k} + \xi ^ {j} \Gamma _ {jk} ^ {i} \right ) \ dx ^ {k} e _ {i} , $$

is called the covariant differential of the field $ X $ with respect to the given affine connection. Here

$$ \nabla _ {k} \xi ^ {i} \ = \ \frac{\partial \xi ^ i}{\partial x ^ k} + \xi ^ {j} \Gamma _ {jk} ^ {i} $$

form a tensor field, called the covariant derivative of the field $ X = \xi ^ {i} e _ {i} $. If a second vector field $ Y = \eta ^ {k} e _ {k} $ is given, the covariant derivative of the field $ X $ in the direction of $ Y $ is defined as

$$ \nabla _ {Y} X \ = \ \eta ^ {k} \nabla _ {k} \xi ^ {i} e _ {i} , $$

which may also be defined with respect to an arbitrary field of frames by the formula

$$ \omega ^ {i} ( \nabla _ {Y} X ) \ = \ Y \omega ^ {i} (X) + \omega _ {k} ^ {i} (Y) \omega ^ {k} (X). $$

An affine connection on $ M $ may also be defined as a bilinear operator $ \nabla $ which assigns a vector field $ \nabla _ {Y} X $ to each two vector fields $ X $ and $ Y $, and which possesses the properties:

$$ \nabla _ {Y} ( f X ) \ = \ ( Y f ) X + f \nabla _ {Y} X , \ \ \nabla _ {fY} X \ = \ f \nabla _ {Y} X , $$

where $ f $ is a smooth function on $ M $. The relation between these definitions is established by the formula $ \nabla _ {e _ k} e _ {j} = \Gamma _ {jk} ^ {i} e _ {i} $ where $ \{ e _ {i} \} $ is the field of frames. The fields of the torsion tensor and curvature tensor

$$ S ( X ,\ Y ) \ = \ S _ {jk} ^ {i} \xi ^ {j} \eta ^ {k} e _ {i} \ = \ \Omega ^ {i} ( X ,\ Y ) e _ {i} , $$

$$ R ( X ,\ Y ) Z \ = \ ( R _ {jkl} ^ {i} \xi ^ {k} \eta ^ {l} ) \xi ^ {j} e _ {i} \ = \ \Omega _ {j} ^ {i} ( X ,\ Y ) \omega ^ {j} ( Z ) e _ {i} $$

are defined by the formulas:

$$ S ( X ,\ Y ) \ = \ \nabla _ {X} Y - \nabla _ {Y} X - [ X ,\ Y ], $$

$$ R ( X ,\ Y ) \ = \ \nabla _ {X} \nabla _ {Y} Z - \nabla _ {Y} \nabla _ {X} Z - \nabla _ {[ X ,\ Y ]} Z . $$

A vector field $ X $ is said to be parallel along the curve $ L $ if $ \nabla _ {\dot{x} (t)} X _ {x(t)} = 0 $ holds identically with respect to $ t $, i.e. if, along $ L $,

$$ d \xi ^ {i} + \xi ^ {j} \omega _ {j} ^ {i} \ = \ 0 . $$

Parallel vector fields are used to effect parallel displacement of vectors (and, generally, of tensors) in an affine connection, representing a linear mapping of the tangent vector spaces $ T _ {x _ t} (M) \rightarrow T _ {x _ 0} (M) $, defined by the mapping $ ( A _ {n} ) _ {x _ t} \rightarrow ( A _ {n} ) _ {x _ 0} $. In this sense any affine connection generates a linear connection on $ M $.

A curve $ L $ is called a geodesic line in a given affine connection if its development is a straight line; in other words, if, by a suitable parametrization, its tangent vector field $ \dot{x} (t) $ is parallel to it. Geodesic lines are defined with respect to a local coordinate system by the system

$$ \frac{d ^ {2} x ^ i}{d t ^ 2} + \Gamma _ {jk} ^ {i} \frac{d x ^ j}{dt} \frac{d x ^ k}{dt} \ = \ 0. $$

Through each point, in each direction passes one geodesic line.

There is a one-to-one correspondence between affine connections on $ M $ and connections in principal fibre bundles of free affine frames in $ (A _ {n} ) _ {x} ,\ x \in M $, generated by them. To closed curves with origin and end at $ x $ there correspond affine transformations $ ( A _ {n} ) _ {x} \rightarrow ( A _ {n} ) _ {x} $, which form the non-homogeneous holonomy group of the given affine connection. The corresponding linear automorphisms $ T _ {x} (M) \rightarrow T _ {x} (M) $ form the homogeneous holonomy group. In accordance with the holonomy theorem, the Lie algebras of these groups are defined by the $ 2 $- forms of torsion $ \Omega ^ {i} $ and curvature $ \Omega _ {j} ^ {i} $. The Bianchi identities apply to the latter:

$$ d \Omega ^ {i} \ = \ \Omega _ {j} ^ {i} \wedge \omega ^ {i} - \omega _ {j} ^ {i} \wedge \Omega ^ {j} ,\ \ d \Omega _ {j} ^ {i} \ = \ \Omega _ {k} ^ {i} \wedge \omega _ {j} ^ {k} - \omega _ {k} ^ {i} \wedge \Omega _ {j} ^ {k} . $$

In particular, for torsion-free affine connections, when $ \Omega ^ {i} = 0 $, these identities reduce to the following:

$$ R _ {jkl} ^ {i} + R _ {klj} ^ {i} + R _ {ljk} ^ {i} \ = \ 0 ,\ \ \nabla _ {m} R _ {jkl} ^ {i} + \nabla _ {k} R _ {jlm} ^ {i} + \nabla _ {l} R _ {jmk} ^ {i} \ = \ 0 . $$

The concept of an affine connection arose in 1917 in Riemannian geometry (in the form of the Levi-Civita connection); it found an independent meaning in 1918–1924 owing to work of H. Weyl [1] and E. Cartan .

References

[1] H. Weyl, "Raum, Zeit, Materie" , Springer (1923)
[2a] E. Cartan, "Sur les variétés a connexion affine et la théorie de la relativité généralisée (première partie)" Ann. Sci. École Norm. Sup. , 40 (1923) pp. 325–412
[2b] E. Cartan, "Sur les variétés a connexion affine et la théorie de la relativité généralisée (première partie suite)" Ann. Sci. École Norm. Sup. , 41 (1924) pp. 1–25
[2c] E. Cartan, "Sur les variétés a connexion affine et la théorie de la relativité généralisée (deuxième partie)" Ann. Sci. École Norm. Sup. , 42 (1925) pp. 17–88
[3a] E. Cartan, "Sur les variétés à connexion projective" Bull. Soc. Math. France , 52 (1924) pp. 205–241
[3b] E. Cartan, "Sur les espaces à connexion conforme" Ann. Soc. Polon. Math. , 2 (1923) pp. 171–221
[4] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[5] M.M. Postnikov, "The variational theory of geodesics" , Saunders (1967) (Translated from Russian)

Comments

Instead of the articles [3a], [3b], one may consult [a1]. Useful additional up-to-date references in English are [a2] and [a3].

References

[a1] A. Lichnerowicz, "Théorie globale des connexions et des groupes d'holonomie" , Cremonese (1955)
[a2] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)
[a3] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)
How to Cite This Entry:
Affine connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_connection&oldid=44383
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article