# Connection

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on a fibre bundle

A differential-geometric structure on a smooth fibre bundle with a Lie structure group that generalizes 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 along $L$. The parallel displacements along all possible closed curves $L ( x _ {0} , x _ {0} )$ form the 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 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, which determines it uniquely and can be the basis for another definition. An important characteristic of a linear connection is the 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  on parallel displacement of a vector in Riemannian geometry. The notion of an affine connection was introduced by H. Weyl in 1918. In the 1920s E. Cartan (see ) investigated projective and conformal connections (cf. Projective connection; Conformal connection). In 1926 he gave the general concept of a "non-holonomic space with a fundamental group" (see 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 , the first notes of G.F. Laptev, which disclosed new approaches, especially analytic ones, and the work of C. Ehresmann  that laid the foundation of the modern global theory of connections. See also Weyl connection; Linear connection; Riemannian connection; Symplectic connection; Hermitian connection.

How to Cite This Entry:
Connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connection&oldid=46475
This article was adapted from an original article by Ãœ. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article