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Differential-geometric structures (cf. [[Differential-geometric structure|Differential-geometric structure]]) on a smooth [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c0251801.png" /> that are connections (cf. [[Connection|Connection]]) on smooth fibre bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c0251802.png" /> with homogeneous spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c0251803.png" /> of the same dimension as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c0251804.png" /> as typical fibres over the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c0251805.png" />. Depending on the choice of the homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c0251806.png" /> one obtains, for example, affine, projective, conformal, etc., connections on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c0251807.png" /> (cf. [[Affine connection|Affine connection]]; [[Conformal connection|Conformal connection]]; [[Projective connection|Projective connection]]). The general notion of a connection on a manifold was introduced by E. Cartan [[#References|[1]]], who called a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c0251808.png" /> with a connection defined on it a  "non-holonomic space with a fundamental groupnon-holonomic space with a fundamental group" .
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The modern definition of a connection on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c0251809.png" /> is based on the concept of a smooth fibre bundle over the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518010.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518011.png" /> be a [[Homogeneous space|homogeneous space]] of the same dimension as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518012.png" /> (for example, an affine space, a projective space, etc.). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518013.png" /> be a smooth locally trivial fibration with typical fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518014.png" /> and suppose that in this fibration there is fixed a smooth section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518015.png" />, that is, a smooth mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518017.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518018.png" />. The last condition ensures that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518019.png" /> is a diffeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518020.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518021.png" />, and therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518023.png" /> can be identified, if desired. In other words, to each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518024.png" /> there is associated a copy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518025.png" /> of the homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518026.png" /> of the same dimension as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518027.png" /> (that is, the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518028.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518029.png" />) with a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518030.png" /> that can be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518031.png" />.
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A connection on a manifold is a special case of the more general concept of a connection; it can be defined independently as follows. Suppose that for each piecewise-smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518032.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518033.png" /> there is an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518034.png" /> of the tangent homogeneous spaces at the end points of the curve (for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518035.png" /> is an affine or projective space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518036.png" /> is, respectively, an affine or projective mapping). In addition, suppose that
+
Differential-geometric structures (cf. [[Differential-geometric structure|Differential-geometric structure]]) on a smooth [[Manifold|manifold]]  $  M $
 +
that are connections (cf. [[Connection|Connection]]) on smooth fibre bundles  $  E $
 +
with homogeneous spaces  $  G / H $
 +
of the same dimension as  $  M $
 +
as typical fibres over the base  $  M $.
 +
Depending on the choice of the homogeneous space  $  G / H $
 +
one obtains, for example, affine, projective, conformal, etc., connections on  $  M $(
 +
cf. [[Affine connection|Affine connection]]; [[Conformal connection|Conformal connection]]; [[Projective connection|Projective connection]]). The general notion of a connection on a manifold was introduced by E. Cartan [[#References|[1]]], who called a manifold  $  M $
 +
with a connection defined on it a  "non-holonomic space with a fundamental groupnon-holonomic space with a fundamental group" .
  
1) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518039.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518040.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518042.png" />;
+
The modern definition of a connection on a manifold  $  M $
 +
is based on the concept of a smooth fibre bundle over the base  $  M $.
 +
Let  $  F = G / H $
 +
be a [[Homogeneous space|homogeneous space]] of the same dimension as  $  M $(
 +
for example, an affine space, a projective space, etc.). Let  $  p :  E \rightarrow M $
 +
be a smooth locally trivial fibration with typical fibre  $  F $
 +
and suppose that in this fibration there is fixed a smooth section  $  s $,
 +
that is, a smooth mapping  $  s : M \rightarrow E $
 +
such that  $  p ( s ( x) ) = x $
 +
for every  $  x \in M $.  
 +
The last condition ensures that  $  s $
 +
is a diffeomorphism of  $  M $
 +
onto  $  s ( M) $,  
 +
and therefore  $  M $
 +
and  $  s ( M) $
 +
can be identified, if desired. In other words, to each point  $  x \in M $
 +
there is associated a copy  $  F _ {x} $
 +
of the homogeneous space  $  F $
 +
of the same dimension as  $  M $(
 +
that is, the fibre of  $  p : E \rightarrow M $
 +
over  $  x $)
 +
with a fixed point  $  s ( x) $
 +
that can be identified with  $  x $.
  
2) for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518043.png" /> and for each tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518044.png" /> the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518046.png" /> denotes the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518047.png" /> under the parametrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518048.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518049.png" /> with tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518050.png" />, tends to the identity isomorphism as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518051.png" />, and its deviation from the latter depends in its principal part only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518053.png" />, and this dependence is smooth.
+
A connection on a manifold is a special case of the more general concept of a connection; it can be defined independently as follows. Suppose that for each piecewise-smooth curve  $  L ( x _ {0} , x _ {1} ) $
 +
on a manifold  $  M $
 +
there is an isomorphism $  \Gamma L : F _ {x _ {1}  } \rightarrow F _ {x _ {0}  } $
 +
of the tangent homogeneous spaces at the end points of the curve (for example, if  $  F $
 +
is an affine or projective space, then  $  \Gamma L $
 +
is, respectively, an affine or projective mapping). In addition, suppose that
  
In this case it is said that a connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518054.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518055.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518056.png" />; the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518057.png" /> is called the [[Parallel displacement(2)|parallel displacement]] along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518058.png" />. For each curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518059.png" /> its evolute is defined, that is, the curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518060.png" /> that consists of the image of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518061.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518062.png" /> under parallel displacement along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518063.png" />. It follows from 2) that curves with common tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518064.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518065.png" /> have evolutes with common tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518066.png" /> that depends smoothly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518068.png" />. A consequence of this is that for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518069.png" /> there is a mapping
+
1) for  $  L ( x _ {0} , x _ {1} ) $,
 +
$  L  ^  \prime  ( x _ {1} , x _ {2} ) $,  
 +
$  L  ^ {-} 1 ( x _ {1} , x _ {0} ) $,
 +
and  $  L L  ^  \prime  ( x _ {0} , x _ {2} ) $
 +
one has  $  \Gamma L  ^ {-} 1 = ( \Gamma L )  ^ {-} 1 $,
 +
$  \Gamma ( L L  ^  \prime  ) = ( \Gamma L) ( \Gamma L  ^  \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/c/c025/c025180/c02518070.png" /></td> </tr></table>
+
2) for each point  $  x \in M $
 +
and for each tangent vector  $  X _ {x} \in T _ {x} ( M) $
 +
the isomorphism  $  \Gamma L _ {t} : F _ {x _ {t}  } \rightarrow F _ {x} $,
 +
where  $  L _ {t} $
 +
denotes the image of  $  [ 0 , t ] $
 +
under the parametrization  $  \lambda : [ 0 , 1 ] \rightarrow L ( x , x _ {1} ) $
 +
of  $  L $
 +
with tangent vector  $  X $,
 +
tends to the identity isomorphism as  $  t \rightarrow 0 $,
 +
and its deviation from the latter depends in its principal part only on  $  x $
 +
and  $  X $,
 +
and this dependence is smooth.
 +
 
 +
In this case it is said that a connection  $  \Gamma $
 +
of type  $  F $
 +
is defined on  $  M $;  
 +
the isomorphism  $  \Gamma L $
 +
is called the [[Parallel displacement(2)|parallel displacement]] along  $  L $.
 +
For each curve  $  L ( x , x _ {1} ) \in M $
 +
its evolute is defined, that is, the curve in  $  F _ {x} $
 +
that consists of the image of the points  $  x _ {t} $
 +
of  $  L $
 +
under parallel displacement along  $  L _ {t} $.  
 +
It follows from 2) that curves with common tangent vector  $  X $
 +
at a point  $  x $
 +
have evolutes with common tangent vector  $  Y $
 +
that depends smoothly on  $  x $
 +
and  $  X $.  
 +
A consequence of this is that for each point  $  x $
 +
there is a mapping
 +
 
 +
$$
 +
f _ {x} :  T _ {x} ( M)  \rightarrow  T _ {s ( x) }
 +
( F _ {x} ) .
 +
$$
  
 
The connections on a manifold that have been studied most are linear connections, which have the following additional property:
 
The connections on a manifold that have been studied most are linear connections, which have the following additional property:
  
3) the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518071.png" /> in the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518072.png" /> of the structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518073.png" /> that defines the principal part of the deviation of the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518074.png" /> from the identity isomorphism as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518075.png" /> relative to a certain field of frames, depends linearly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518076.png" />.
+
3) the element $  \omega ( x) $
 +
in the Lie algebra $  \mathfrak g $
 +
of the structure group $  G $
 +
that defines the principal part of the deviation of the isomorphism $  \Gamma L _ {t} $
 +
from the identity isomorphism as $  t \rightarrow 0 $
 +
relative to a certain field of frames, depends linearly on $  X $.
  
In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518077.png" /> is a linear mapping. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518078.png" /> is an isomorphism for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518079.png" />, then one speaks about a non-degenerate connection on a manifold, or about a Cartan connection; in this case the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518080.png" /> is also treated as a glueing of the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518081.png" /> to the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518082.png" /> (along a given section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518083.png" />). A Cartan connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518084.png" /> is called complete if for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518085.png" />, any smooth curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518086.png" /> that begins at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518087.png" /> is the evolute of a curve on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518088.png" />.
+
In this case $  f _ {x} $
 +
is a linear mapping. If $  f _ {x} $
 +
is an isomorphism for any point $  x $,  
 +
then one speaks about a non-degenerate connection on a manifold, or about a Cartan connection; in this case the isomorphism $  f _ {x}  ^ {-} 1 $
 +
is also treated as a glueing of the fibration $  p : E \rightarrow M $
 +
to the base $  M $(
 +
along a given section $  s $).  
 +
A Cartan connection on $  M $
 +
is called complete if for each point $  x $,  
 +
any smooth curve in $  F _ {x} $
 +
that begins at $  x $
 +
is the evolute of a curve on $  M $.
  
There is another point of view of the general theory of connections, where a linear connection in the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518089.png" /> is defined by using a [[Horizontal distribution|horizontal distribution]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518090.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518091.png" />. Then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518092.png" /> is the composite of an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518093.png" /> that maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518094.png" /> into the corresponding tangent vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518095.png" />, followed by a projection of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518096.png" /> onto the second direct summand. Hence it follows that a connection is non-degenerate if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518097.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518098.png" />. To <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c02518099.png" /> all concepts and results developed in the general theory of connections can be applied. Such are, e.g., the [[Holonomy group|holonomy group]], the [[Curvature form|curvature form]], the holonomy theorem, etc. The additional structure of a fibre bundle over the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180100.png" /> enables one, however, to introduce certain more special concepts. Apart from evolutes, the most most important of these is the concept of the [[Torsion form|torsion form]] of a connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180101.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180102.png" />.
+
There is another point of view of the general theory of connections, where a linear connection in the fibration $  p : E \rightarrow M $
 +
is defined by using a [[Horizontal distribution|horizontal distribution]] $  \Delta $
 +
on $  E $.  
 +
Then the mapping $  f _ {x} $
 +
is the composite of an isomorphism $  s  ^ {*} $
 +
that maps $  X $
 +
into the corresponding tangent vector to $  s ( M) $,  
 +
followed by a projection of the space $  T _ {s ( x) }  ( E) = \Delta _ {s ( x) }  \oplus T _ {s ( x) }  ( F _ {x} ) $
 +
onto the second direct summand. Hence it follows that a connection is non-degenerate if and only if $  \Delta _ {s ( x) }  \cap T _ {s ( x) }  ( s ( M) ) = \{ 0 \} $
 +
for any $  x \in M $.  
 +
To $  M $
 +
all concepts and results developed in the general theory of connections can be applied. Such are, e.g., the [[Holonomy group|holonomy group]], the [[Curvature form|curvature form]], the holonomy theorem, etc. The additional structure of a fibre bundle over the manifold $  M $
 +
enables one, however, to introduce certain more special concepts. Apart from evolutes, the most most important of these is the concept of the [[Torsion form|torsion form]] of a connection on $  M $
 +
at $  x $.
  
The Cartan connections in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180103.png" /> is a homogeneous [[Reductive space|reductive space]] (that is, when there is a direct decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180104.png" /> with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180105.png" />) occupy a special position in the theory of connections on a manifold. In this case the curvature form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180106.png" /> splits into two independent objects: its component in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180107.png" /> generates the torsion form, and the component in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180108.png" /> generates the curvature form. The best-known example here is an affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180109.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180110.png" /> is an affine space of the same dimension as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180111.png" />.
+
The Cartan connections in the case when $  F = G / H $
 +
is a homogeneous [[Reductive space|reductive space]] (that is, when there is a direct decomposition $  \mathfrak g = \mathfrak k + \mathfrak m $
 +
with the property $  [ \mathfrak h \mathfrak m ] \subset  \mathfrak m $)  
 +
occupy a special position in the theory of connections on a manifold. In this case the curvature form $  \Omega $
 +
splits into two independent objects: its component in $  \mathfrak m $
 +
generates the torsion form, and the component in $  \mathfrak h $
 +
generates the curvature form. The best-known example here is an affine connection on $  M $
 +
for which $  F $
 +
is an affine space of the same dimension as $  M $.
  
A reductive space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180112.png" /> has an invariant affine connection. More generally, if there is an invariant affine or projective connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180113.png" />, then the geodesic lines (cf. [[Geodesic line|Geodesic line]]) of a connection of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180114.png" /> are defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180115.png" /> as those lines possessing evolutes which are geodesic lines of the given invariant connection.
+
A reductive space $  F $
 +
has an invariant affine connection. More generally, if there is an invariant affine or projective connection on $  F $,  
 +
then the geodesic lines (cf. [[Geodesic line|Geodesic line]]) of a connection of type $  F $
 +
are defined on $  M $
 +
as those lines possessing evolutes which are geodesic lines of the given invariant connection.
  
 
====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">  Ch. Ehresmann,  "Les connexions infinitésimal dans une espace fibré différentiable" , ''Colloq. de Topologie Bruxelles, 1950'' , G. Thone &amp; Masson  (1951)  pp. 29–55</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Kobayashi,  "On connections of Cartan"  ''Canad. J. Math.'' , '''8''' :  2  (1956)  pp. 145–156</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Y.H. Clifton,  "On the completeness of Cartan connections"  ''J. Math. Mech.'' , '''16''' :  6  (1966)  pp. 569–576</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">  Ch. Ehresmann,  "Les connexions infinitésimal dans une espace fibré différentiable" , ''Colloq. de Topologie Bruxelles, 1950'' , G. Thone &amp; Masson  (1951)  pp. 29–55</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Kobayashi,  "On connections of Cartan"  ''Canad. J. Math.'' , '''8''' :  2  (1956)  pp. 145–156</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Y.H. Clifton,  "On the completeness of Cartan connections"  ''J. Math. Mech.'' , '''16''' :  6  (1966)  pp. 569–576</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180116.png" /> be a trivial [[Vector bundle|vector bundle]]. The principal part of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180117.png" /> is the component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180118.png" />. Similarly, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180119.png" /> is a bundle homomorphism (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180120.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180121.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025180/c025180122.png" />, is its principal part. See also the editorial comments to [[Connection|Connection]].
+
Let $  E = U \times F $
 +
be a trivial [[Vector bundle|vector bundle]]. The principal part of an element $  e = ( u , f  ) \in E $
 +
is the component $  f $.  
 +
Similarly, if $  \phi : E \rightarrow E $
 +
is a bundle homomorphism ( $  ( u , f  ) \mapsto ( u , g( u ) f  ) $),  
 +
then $  g( u ) $,  
 +
or $  u \mapsto g( u ) $,  
 +
is its principal part. See also the editorial comments to [[Connection|Connection]].

Revision as of 17:46, 4 June 2020


Differential-geometric structures (cf. Differential-geometric structure) on a smooth manifold $ M $ that are connections (cf. Connection) on smooth fibre bundles $ E $ with homogeneous spaces $ G / H $ of the same dimension as $ M $ as typical fibres over the base $ M $. Depending on the choice of the homogeneous space $ G / H $ one obtains, for example, affine, projective, conformal, etc., connections on $ M $( cf. Affine connection; Conformal connection; Projective connection). The general notion of a connection on a manifold was introduced by E. Cartan [1], who called a manifold $ M $ with a connection defined on it a "non-holonomic space with a fundamental groupnon-holonomic space with a fundamental group" .

The modern definition of a connection on a manifold $ M $ is based on the concept of a smooth fibre bundle over the base $ M $. Let $ F = G / H $ be a homogeneous space of the same dimension as $ M $( for example, an affine space, a projective space, etc.). Let $ p : E \rightarrow M $ be a smooth locally trivial fibration with typical fibre $ F $ and suppose that in this fibration there is fixed a smooth section $ s $, that is, a smooth mapping $ s : M \rightarrow E $ such that $ p ( s ( x) ) = x $ for every $ x \in M $. The last condition ensures that $ s $ is a diffeomorphism of $ M $ onto $ s ( M) $, and therefore $ M $ and $ s ( M) $ can be identified, if desired. In other words, to each point $ x \in M $ there is associated a copy $ F _ {x} $ of the homogeneous space $ F $ of the same dimension as $ M $( that is, the fibre of $ p : E \rightarrow M $ over $ x $) with a fixed point $ s ( x) $ that can be identified with $ x $.

A connection on a manifold is a special case of the more general concept of a connection; it can be defined independently as follows. Suppose that for each piecewise-smooth curve $ L ( x _ {0} , x _ {1} ) $ on a manifold $ M $ there is an isomorphism $ \Gamma L : F _ {x _ {1} } \rightarrow F _ {x _ {0} } $ of the tangent homogeneous spaces at the end points of the curve (for example, if $ F $ is an affine or projective space, then $ \Gamma L $ is, respectively, an affine or projective mapping). In addition, suppose that

1) for $ L ( x _ {0} , x _ {1} ) $, $ L ^ \prime ( x _ {1} , x _ {2} ) $, $ L ^ {-} 1 ( x _ {1} , x _ {0} ) $, and $ L L ^ \prime ( x _ {0} , x _ {2} ) $ one has $ \Gamma L ^ {-} 1 = ( \Gamma L ) ^ {-} 1 $, $ \Gamma ( L L ^ \prime ) = ( \Gamma L) ( \Gamma L ^ \prime ) $;

2) for each point $ x \in M $ and for each tangent vector $ X _ {x} \in T _ {x} ( M) $ the isomorphism $ \Gamma L _ {t} : F _ {x _ {t} } \rightarrow F _ {x} $, where $ L _ {t} $ denotes the image of $ [ 0 , t ] $ under the parametrization $ \lambda : [ 0 , 1 ] \rightarrow L ( x , x _ {1} ) $ of $ L $ with tangent vector $ X $, tends to the identity isomorphism as $ t \rightarrow 0 $, and its deviation from the latter depends in its principal part only on $ x $ and $ X $, and this dependence is smooth.

In this case it is said that a connection $ \Gamma $ of type $ F $ is defined on $ M $; the isomorphism $ \Gamma L $ is called the parallel displacement along $ L $. For each curve $ L ( x , x _ {1} ) \in M $ its evolute is defined, that is, the curve in $ F _ {x} $ that consists of the image of the points $ x _ {t} $ of $ L $ under parallel displacement along $ L _ {t} $. It follows from 2) that curves with common tangent vector $ X $ at a point $ x $ have evolutes with common tangent vector $ Y $ that depends smoothly on $ x $ and $ X $. A consequence of this is that for each point $ x $ there is a mapping

$$ f _ {x} : T _ {x} ( M) \rightarrow T _ {s ( x) } ( F _ {x} ) . $$

The connections on a manifold that have been studied most are linear connections, which have the following additional property:

3) the element $ \omega ( x) $ in the Lie algebra $ \mathfrak g $ of the structure group $ G $ that defines the principal part of the deviation of the isomorphism $ \Gamma L _ {t} $ from the identity isomorphism as $ t \rightarrow 0 $ relative to a certain field of frames, depends linearly on $ X $.

In this case $ f _ {x} $ is a linear mapping. If $ f _ {x} $ is an isomorphism for any point $ x $, then one speaks about a non-degenerate connection on a manifold, or about a Cartan connection; in this case the isomorphism $ f _ {x} ^ {-} 1 $ is also treated as a glueing of the fibration $ p : E \rightarrow M $ to the base $ M $( along a given section $ s $). A Cartan connection on $ M $ is called complete if for each point $ x $, any smooth curve in $ F _ {x} $ that begins at $ x $ is the evolute of a curve on $ M $.

There is another point of view of the general theory of connections, where a linear connection in the fibration $ p : E \rightarrow M $ is defined by using a horizontal distribution $ \Delta $ on $ E $. Then the mapping $ f _ {x} $ is the composite of an isomorphism $ s ^ {*} $ that maps $ X $ into the corresponding tangent vector to $ s ( M) $, followed by a projection of the space $ T _ {s ( x) } ( E) = \Delta _ {s ( x) } \oplus T _ {s ( x) } ( F _ {x} ) $ onto the second direct summand. Hence it follows that a connection is non-degenerate if and only if $ \Delta _ {s ( x) } \cap T _ {s ( x) } ( s ( M) ) = \{ 0 \} $ for any $ x \in M $. To $ M $ all concepts and results developed in the general theory of connections can be applied. Such are, e.g., the holonomy group, the curvature form, the holonomy theorem, etc. The additional structure of a fibre bundle over the manifold $ M $ enables one, however, to introduce certain more special concepts. Apart from evolutes, the most most important of these is the concept of the torsion form of a connection on $ M $ at $ x $.

The Cartan connections in the case when $ F = G / H $ is a homogeneous reductive space (that is, when there is a direct decomposition $ \mathfrak g = \mathfrak k + \mathfrak m $ with the property $ [ \mathfrak h \mathfrak m ] \subset \mathfrak m $) occupy a special position in the theory of connections on a manifold. In this case the curvature form $ \Omega $ splits into two independent objects: its component in $ \mathfrak m $ generates the torsion form, and the component in $ \mathfrak h $ generates the curvature form. The best-known example here is an affine connection on $ M $ for which $ F $ is an affine space of the same dimension as $ M $.

A reductive space $ F $ has an invariant affine connection. More generally, if there is an invariant affine or projective connection on $ F $, then the geodesic lines (cf. Geodesic line) of a connection of type $ F $ are defined on $ M $ as those lines possessing evolutes which are geodesic lines of the given invariant connection.

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] Ch. Ehresmann, "Les connexions infinitésimal dans une espace fibré différentiable" , Colloq. de Topologie Bruxelles, 1950 , G. Thone & Masson (1951) pp. 29–55
[4] S. Kobayashi, "On connections of Cartan" Canad. J. Math. , 8 : 2 (1956) pp. 145–156
[5] Y.H. Clifton, "On the completeness of Cartan connections" J. Math. Mech. , 16 : 6 (1966) pp. 569–576

Comments

Let $ E = U \times F $ be a trivial vector bundle. The principal part of an element $ e = ( u , f ) \in E $ is the component $ f $. Similarly, if $ \phi : E \rightarrow E $ is a bundle homomorphism ( $ ( u , f ) \mapsto ( u , g( u ) f ) $), then $ g( u ) $, or $ u \mapsto g( u ) $, is its principal part. See also the editorial comments to Connection.

How to Cite This Entry:
Connections on a manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connections_on_a_manifold&oldid=13655
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article