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''on a fibre bundle''

A [[Differential-geometric structure|differential-geometric structure]] on a smooth fibre bundle with a Lie structure group that generalizes [[Connections on a manifold|connections on a manifold]], in particular, for example, the [[Levi-Civita connection]] in Riemannian geometry. Let  $  p :  E \rightarrow B $
be a smooth locally trivial fibration with typical fibre  $  F $
on which a Lie group  $  G $
acts effectively and smoothly. A connection on this fibre bundle is a mapping of the category of piecewise-smooth curves in the base  $  B $
into the category of diffeomorphisms of the fibres that associates with a curve  $  L = L ( x _ {0} , x _ {1} ) $(
with initial point  $  x _ {0} $
and end point  $  x _ {1} $) 
a diffeomorphism  $  \Gamma L :  p  ^ {-1} ( x _ {1} ) \rightarrow p  ^ {-1} ( x _ {0} ) $
satisfying the following axioms:

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 an arbitrary trivializing diffeomorphism  $  \phi :  U \times F \rightarrow p  ^ {-1} ( U) $
and for an  $  L ( x _ {0} , x _ {1} ) \subset  U $, 
the diffeomorphism  $  \phi _ {x _ {0}  }  ^ {-1} ( \Gamma L ) \phi _ {x _ {1}  } :  ( x _ {1} , F ) \rightarrow ( x _ {0} , F ) $, 
where  $  \phi _ {x} = \phi \mid  _ {( x , F ) }  $, 
is defined by the action of some element  $  g _  \phi   ^ {L} \in G $;

3) for an arbitrary piecewise-smooth parametrization  $  \lambda :  [ 0 , 1 ] \rightarrow L ( x _ {0} , x _ {1} ) \subset  U $, 
the mapping  $  t \mapsto g _  \phi  ^ {L _ {t} } $, 
where  $  L _ {t} $
is the image of  $  [ 0 , t ] $
under  $  \lambda $, 
defines a piecewise-smooth curve in  $  G $
that starts from the unit element  $  e = g _  \phi  ^ {L _ {0} } $; 
moreover,  $  \lambda , \lambda  ^  \prime  :  [ 0 , 1 ] \rightarrow L  ^  \prime  ( x _ {0} , x _ {1} ) \subset  U $
with a common non-zero tangent vector  $  X \in T _ {x _ {0}  } B $
define paths in  $  G $
with a common tangent vector  $  \theta _  \phi  ( x _ {0} , X ) \in T _ {e} ( G) = g $
that depends smoothly on  $  x _ {0} $
and  $  X $.

The diffeomorphism  $  \Gamma L $
is called the [[Parallel displacement(2)|parallel displacement]] along  $  L $. 
The parallel displacements along all possible closed curves  $  L ( x _ {0} , x _ {0} ) $
form the [[Holonomy group|holonomy group]] of the connection  $  \Gamma $
at  $  x _ {0} $; 
this group is isomorphic to a Lie subgroup of  $  G $
that does not depend on  $  x _ {0} $. 
A curve  $  \Lambda ( y _ {0} , y _ {1} ) $
on  $  E $
is said to be horizontal for  $  \Gamma $
if  $  \Gamma ( p \Lambda _ {t} ) y _ {t} = y _ {0} $
for any  $  t \in [ 0 , 1 ] $
and some piecewise-smooth parametrization of it. If  $  L ( x _ {0} , x _ {1} ) $
and  $  y _ {0} \in p  ^ {-1} ( x _ {0} ) $
are given, then there always exists a unique horizontal curve  $  \Lambda ( y _ {0} , y _ {1} ) $, 
called the horizontal lift of the curve  $  L $, 
such that  $  p \Lambda = L $; 
it consists of the points  $  \Gamma L _ {t}  ^ {-1} y _ {0} $. 
The set of horizontal lifts of all curves  $  L $
in  $  B $
determines the connection  $  \Gamma $
uniquely:  $  \Gamma L $
maps the end points of all lifted curves of  $  L $
into the initial points.

A connection is called linear if  $  \theta _  \phi  ( x , X ) $
depends linearly on  $  X $
for any  $  \phi $
and  $  x $, 
or equivalently, if for any  $  y \in E $
the tangent vectors of the horizontal curves beginning at  $  y $
form a vector subspace  $  \Delta _ {y} $
of  $  T _ {t} ( E) $, 
called the horizontal subspace. Here  $  T _ {y} ( E) = \Delta _ {y} \oplus T _ {y} ( F _ {y} ) $, 
where  $  F _ {y} $
is the fibre through  $  y $, 
that is,  $  F _ {y} = p  ^ {-1} ( p ( y) ) $. 
The smooth distribution  $  \Delta :  y \mapsto \Delta _ {y} $
is called the [[Horizontal distribution|horizontal distribution]] of the linear connection  $  \Gamma $. 
It determines  $  \Gamma $
uniquely: its integral curves are the horizontal lifts.

A fibre bundle  $  E $
is called principal (respectively, a space of homogeneous type), and is denoted by  $  P $(
respectively,  $  Q $), 
if  $  G $
acts simply transitively (respectively, transitively) on  $  F $, 
that is, if for any  $  z , z  ^  \prime  \in F $
there is exactly one (respectively, there is an) element  $  g \in G $
that sends  $  z $
to  $  z  ^  \prime  $. 
Suppose that  $  G $
acts on  $  F $
from the left; then a natural action from the right is defined on  $  P $, 
where  $  g $
defines  $  R _ {g} :  z \mapsto z \circ g $. 
Here  $  Q $
is identified with the quotient manifold  $  P / H $
formed by the orbits  $  y \circ H $, 
where  $  H $
is the stationary subgroup of a point from  $  F = G / H $. 
More generally,  $  E $
can be identified with the quotient manifold  $  ( P \times F ) / G $
of orbits  $  ( y , z ) \circ G $
relative to the action defined by  $  ( y , z ) \circ g = ( y \circ g , g  ^ {-1} \circ z ) $.

A smooth distribution  $  \Delta $
on  $  P $
is a horizontal distribution of some linear connection (which it determines uniquely) if and only if

$$ 
T _ {y} ( P)  =  \Delta _ {y} \oplus T _ {y} ( F _ {y} ) ,\ \ 
R _ {g}  ^ {*} \Delta _ {y}  = \ 
\Delta _ {y} \circ g
$$

for arbitrary  $  y \in P $
and  $  g \in G $. 
All horizontal distributions on  $  Q $(
respectively,  $  P $) 
are the images of such  $  \Delta $
under the canonical projection  $  P \rightarrow Q = P / H $(
respectively, the natural lifts of such  $  \Delta $
to  $  P \times F $
under the canonical projection  $  P \times F \rightarrow E = ( P \times F ) / G $). 
Often a linear connection is defined directly as a distribution with the properties mentioned above. It is known that on each  $  P $, 
and so on every  $  Q $
and  $  E $, 
there is a linear connection.

A linear connection in  $  P $
is usually studied by using its [[Connection form|connection form]], which determines it uniquely and can be the basis for another definition. An important characteristic of a linear connection is the [[Curvature form|curvature form]]; this can be used to compute the Lie algebra of the holonomy group.

The idea of a connection first arose in 1917 in the work of T. Levi-Civita [[#References|[1]]] on parallel displacement of a vector in Riemannian geometry. The notion of an [[Affine connection|affine connection]] was introduced by H. Weyl in 1918. In the 1920s E. Cartan (see [[#References|[3]]]–[[#References|[5]]]) investigated projective and conformal connections (cf. [[Projective connection|Projective connection]]; [[Conformal connection|Conformal connection]]). In 1926 he gave the general concept of a  "non-holonomic space with a fundamental group"  (see [[Connections on a manifold|Connections on a manifold]]), and identified these spaces from the point of view of the general theory of connections. In the 1940s V.V. Vagner developed an even more general concept that is close in spirit (but not in terms of the method) to the modern idea of a connection. 1950 was a decisive year; in it there appeared the survey by Vagner [[#References|[6]]], the first notes of G.F. Laptev, which disclosed new approaches, especially analytic ones, and the work of C. Ehresmann [[#References|[7]]] that laid the foundation of the modern global theory of connections. See also [[Weyl connection|Weyl connection]]; [[Linear connection|Linear connection]]; [[Riemannian connection|Riemannian connection]]; [[Symplectic connection|Symplectic connection]]; [[Hermitian connection|Hermitian connection]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T. Levi-Civita,   "Nozione di parallelismo in una varietà qualunque e consequente specificazione geometrica della curvatura Riemanniana"  ''Rend. Cir. Mat. Palermo'' , '''42'''  (1917)  pp. 173–205</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Weyl,   "Raum, Zeit, Materie" , Springer  (1923)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Cartan,   "Les espaces à connexion conforme"  ''Ann. Soc. Polon. Math.'' , '''2'''  (1924)  pp. 171–221</TD></TR><TR><TD valign="top">[4]</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">[5]</TD> <TD valign="top">  E. Cartan,   "Les groupes d'holonomie des espaces généralisés"  ''Acta Math.'' , '''48'''  (1926)  pp. 1–42</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.V. Vagner,   "Theory of a composite manifold"  ''Trudy Sem. Vektor i Tenzor Anal.'' , '''8'''  (1950)  pp. 11–72  (In Russian)</TD></TR><TR><TD valign="top">[7]</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">[8]</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">[9]</TD> <TD valign="top">  K. Nomizu,   "Lie groups and differential geometry" , Math. Soc. Japan  (1956)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A. Lichnerowicz,   "Global theory of connections and holonomy groups" , Noordhoff  (1955)  (Translated from French)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  Ü.G. Lumiste,   "Connection theory in bundle spaces"  ''J. Soviet Math.'' , '''1'''  (1973)  pp. 363–390  ''Itogi Nauk. Ser. Algebra. Topol. Geom. 1969'' , '''21'''  (1971)  pp. 123–168</TD></TR></table>

====Comments====
Consider a smooth locally trivial fibre bundle  $  p:  E \rightarrow B $. 
A smooth section is a smooth mapping  $  s:  B \rightarrow E $
such that  $  p \circ s =  \mathop{\rm id} $. 
This concept generalizes that of a function  $  B \rightarrow F $(
where  $  F $
is the fibre of  $  p $), 
which is the same as a section of the trivial fibre bundle  $  B \times F \rightarrow B $. 
In several areas of mathematics it is important to consider sections instead of just functions. E.g. in gauge field theory. But then one would also like to have something like the partial derivatives of a section available, i.e. the quantity that describes to first order how  $  s ( b) $
changes as  $  b $
varies (infinitesimally). This requires comparing the fibres of  $  p:  E \rightarrow B $
at neighbouring points, but there is nothing in the concept of a fibre bundle as it stands that allows one to do this. For this some extra structure is needed, and that is provided by the idea of a connection.

It would be simplest if for every two points  $  b, b  ^  \prime  \in B $
one could prescribe an isomorphism  $  \phi _ {b, b  ^  \prime   } :  E _ {b} \rightarrow E _ {b  ^  \prime   } $
in a consistent way, i.e. such that  $  \phi _ {b  ^  \prime   , b  ^ {\prime\prime} } \phi _ {b, b  ^  \prime   } = \phi _ {b, b  ^ {\prime\prime}  } $
for all triples  $  b, b  ^  \prime  , b  ^ {\prime\prime} $. 
Here  $  E _ {b} $, 
the fibre of  $  E $
over  $  b $, 
is of course  $  p  ^ {-1} ( b) $. 
This, however, would make the bundle trivial, and this is in general not possible. The next best thing would be to have for every smooth path  $  L $
from  $  x _ {0} $
to  $  x _ {1} $
an isomorphism  $  \phi _ {L} :  E _ {x _ {0}  } \rightarrow E _ {x _ {1}  } $(
which may depend on the path  $  L $) 
from the fibre at the initial point of the path to the fibre at the final point, subject to certain natural restrictions. This is precisely what a connection is.

There are — at least — three intuitively natural ways of describing a connection.

i) Provide for every smooth path  $  L $
from  $  x _ {0} $
to  $  x _ {1} $
an isomorphism  $  E _ {x _ {1}  } \rightarrow E _ {x _ {0}  } $
subject to the three conditions 1), 2), 3).

ii) For each  $  e \in E $
let  $  VT _ {e} E =  \mathop{\rm Ker} (( T _ {p} ) _ {e} :  T _ {e} E \rightarrow T _ {p ( e) }  B) $
be the kernel of the tangent mapping at  $  e $. 
The subspace  $  VT _ {e} E $
of the tangent space  $  T _ {e} E $
to  $  E $
at  $  e $
is called the vertical tangent subspace to  $  E $
at  $  e $. 
Now for each  $  e \in E $
define a complementary subspace  $  HT _ {e} E $
at  $  e $, 
called the horizontal tangent subspace at  $  e \in E $. 
Thus,  $  T _ {e} E = HT _ {e} E \oplus VT _ {e} E $
and  $  ( Tp) _ {e} $
induces an isomorphism  $  HT _ {e} E \rightarrow T _ {p ( e) }  B $. 
The  $  VT _ {e} E $
are required to vary smoothly with  $  e $. 
In the case of linear connections, cf. above, this is the infinitesimal version of i).

iii) Let  $  E $
be a vector bundle. Then a linear connection can also be specified by giving so to speak the partial derivatives of a section directly (covariant differentiation). This leads to the specification of a bilinear mapping  $  \nabla :  V ( B) \times \Gamma ( E) \rightarrow \Gamma ( E) $, 
where  $  V ( B) $
is the space of vector fields on  $  B $
and  $  \Gamma ( E) $
is the space of sections of  $  p:  E \rightarrow B $, 
with certain properties; cf. [[Linear connection|Linear connection]] for these properties in the case  $  E = TB $. 
One consequence of these properties is that  $  ( \nabla _ {X} s) ( b) $, 
$  X \in V( B) $, 
$  s \in \Gamma ( E) $, 
depends only on  $  X ( b) $
at  $  b $. 
If  $  L :  [ 0 , t ] \rightarrow B $
is a smooth path starting in  $  b $
with tangent vector  $  X ( b) $
at  $  b $, 
then

$$ 
( \nabla _ {X ( b) }  s) ( b)  = \ 
\lim\limits _ {h \rightarrow 0 }  h  ^ {-1}
( \Gamma _ {h} s ( b ( h)) - s ( b) )
$$

where  $  \Gamma _ {h} :  E _ {L ( h) }  \rightarrow E _ {L ( 0) }  = E _ {b} $
is parallel displacement defined by  $  [ 0, h] \rightarrow B $.

An elegant and convenient way to describe a linear connection in the case that  $  E $
is a vector bundle is as follows. Let

$$ 

\begin{array}{ccc}
\pi  ^ {-1} ( U)  &  \mathop \rightarrow \limits ^ { {\widetilde \phi   }}    &\phi ( U) \times \mathbf R  ^ {m}  \\
\downarrow  &{}  &\downarrow  \\
 U  &  \mathop \rightarrow \limits _  \phi     &\phi ( U) \subset  \mathbf R  ^ {n}  \\
\end{array}

$$

be a local chart of  $  B $
and a trivialization of  $  E $. 
Then above  $  \pi  ^ {-1} ( U) $
one has the following local trivialization of  $  TE $:

$$ 

\begin{array}{ccc}
T \pi  ^ {-1} ( U)  &  \mathop \rightarrow \limits ^ { {T \widetilde \phi   }}    &\phi ( U) \times \mathbf R  ^ {m} \times \mathbf R  ^ {n} \times \mathbf R  ^ {m}  \\
\downarrow  &{}  &\downarrow  \\
\pi  ^ {-1} ( U)  &  \mathop \rightarrow \limits _ { {\widetilde \phi   }}    &\phi ( U) \times \mathbf R  ^ {m}  \\
\end{array}

$$

where the right-hand arrow is projection in the first two factors. A linear connection on  $  E $
is now given by a bundle mapping  $  K:  TE \rightarrow E $(
i.e. the diagram

$$ 

\begin{array}{ccc}
TE  & \rightarrow ^ { K }    & E  \\
\downarrow  &{}  &\downarrow P  \\
 E  & \rightarrow _ { P }   & B  \\
\end{array}

$$

is commutative, and  $  K $
is linear in the fibres), such that locally the mapping looks like

$$ 
K _  \phi   = \ 
\widetilde \phi   \circ K \circ
( T \widetilde \phi   )  ^ {-1} : \ 
\phi ( U) \times \mathbf R  ^ {m}
\times \mathbf R  ^ {n} \times \mathbf R  ^ {m}  \rightarrow \ 
\phi ( U) \times \mathbf R  ^ {m} ,
$$

$$ 
( b, \xi , \nu , \eta )  \mapsto  ( b, \eta + \Gamma _  \phi  ( b) ( \nu , \xi )) .
$$

The  $  \Gamma _  \phi  ( b) $
are the Christoffel symbols (relative to the trivialization  $  ( \phi , \widetilde \phi   ) $; 
in case  $  E = TB $, 
$  \widetilde \phi   $
can be taken equal to  $  T \phi $
so that the Christoffel symbols depend only on the chart  $  \phi $).

Given the connection  $  K $, 
the horizontal subspace  $  HT _ {e} E $
is defined by

$$ 
HT _ {e} E  = \ 
 \mathop{\rm Ker} ( K _ {e} :  T _ {e} E  \rightarrow  E _ {p ( e) }  )
$$

and the covariant derivative of a section  $  s:  B \rightarrow E $
along a vector field  $  X:  B \rightarrow TB $
is the section  $  K \circ Ts \circ X:  B \rightarrow TB \rightarrow TE \rightarrow E $.

In the case of infinite-dimensional manifolds and bundles this last notion of a linear connection appears to be the appropriate replacement of the more traditional covariant derivative  $  \nabla :  V ( B) \times \Gamma ( E) \rightarrow \Gamma ( E) $, 
cf. [[#References|[a2]]], Sect. 1.1.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1''' , Interscience  (1963)  pp. Chapt. 4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Klingenberg,   "Lectures on closed geodesics" , Springer  (1979)</TD></TR></table>