In differential geometry, a fibered manifold is surjective submersion of smooth manifolds . Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manifolds provides a general framework of a connection on fiber bundles.
Let be a fibered manifold. A generalized connection on is a section, where is the jet manifold of .[1]
With the above manifold there is the following canonical short exact sequence of vector bundles over :
where and are the tangent bundles of, respectively, is the vertical tangent bundle of, and is the pullback bundle of onto .
A connection on a fibered manifold is defined as a linear bundle morphism
over which splits the exact sequence . A connection always exists.
Sometimes, this connection is called the Ehresmann connection because it yields the horizontal distribution
HY=\Gamma\left(Y x XTX\right)\subsetTY
of and its horizontal decomposition .
At the same time, by an Ehresmann connection also is meant the following construction. Any connection on a fibered manifold yields a horizontal lift of a vector field on onto, but need not defines the similar lift of a path in into . Let
\begin{align}R\supset[,]\nit&\tox(t)\inX\ R\nit&\toy(t)\inY\end{align}
be two smooth paths in and, respectively. Then is called the horizontal lift of if
\pi(y(t))=x(t),
| y(t)\in |
HY, t\inR.
A connection is said to be the Ehresmann connection if, for each path in, there exists its horizontal lift through any point . A fibered manifold is a fiber bundle if and only if it admits such an Ehresmann connection.
Given a fibered manifold, let it be endowed with an atlas of fibered coordinates, and let be a connection on . It yields uniquely the horizontal tangent-valued one-form
on which projects onto the canonical tangent-valued form (tautological one-form or solder form)
| \mu ⊗ \partial | |
| \theta | |
| \mu |
on, and vice versa. With this form, the horizontal splitting reads
\Gamma:\partial\mu\to\partial\mu\rfloor\Gamma=\partial\mu
| i | |
| +\Gamma | |
| \mu\partial |
i.
In particular, the connection in yields the horizontal lift of any vector field on to a projectable vector field
\Gamma
| \mu\left(\partial | |
| \tau=\tau\rfloor\Gamma=\tau | |
| \mu |
| i | |
| +\Gamma | |
| \mu\partial |
i\right)\subsetHY
on .
The horizontal splitting of the exact sequence defines the corresponding splitting of the dual exact sequence
0\toY x XT*X\toT*Y\toV*Y\to0,
where and are the cotangent bundles of, respectively, and is the dual bundle to, called the vertical cotangent bundle. This splitting is given by the vertical-valued form
\Gamma=\left(dyi
| i | |
| -\Gamma | |
| λ |
| λ\right) ⊗ \partial | |
| dx | |
| i, |
which also represents a connection on a fibered manifold.
Treating a connection as a vertical-valued form, one comes to the following important construction. Given a fibered manifold, let be a morphism and the pullback bundle of by . Then any connection on induces the pullback connection
f*\Gamma=\left(dyi-\left(\Gamma\circ\tilde
| i | ||||
| f\right) | ||||
|
| \mu\right) ⊗ \partial | |
| dx' | |
| i |
on .
Let be the jet manifold of sections of a fibered manifold, with coordinates . Due to the canonical imbedding
| 1Y\to | |
| J | |
| Y |
\left(Y x XT*X\right) ⊗ YTY,
| i | |
| \left(y | |
| \mu\right)\to |
dx\mu ⊗ \left(\partial\mu+
| i | |
| y | |
| \mu\partial |
i\right),
any connection on a fibered manifold is represented by a global section
\Gamma:Y\toJ1Y,
| i, | |
| y | |
| λ |
of the jet bundle, and vice versa. It is an affine bundle modelled on a vector bundle
There are the following corollaries of this fact.
Given the connection on a fibered manifold, its curvature is defined as the Nijenhuis differential
\begin{align} R&=\tfrac12d\Gamma\Gamma\\&=\tfrac12[\Gamma,\Gamma]FN\\&=\tfrac12
| i | |
| R | |
| λ\mu |
dxλ\wedge
| \mu ⊗ \partial | |
| dx | |
| i, |
| i | |
| \\ R | |
| λ\mu |
&=\partialλ\Gamma
| i | |
| \mu |
-\partial\mu\Gamma
| i | |
| λ |
+
| j\partial | |
| \Gamma | |
| j |
| i | |
| \Gamma | |
| \mu |
-
| j\partial | |
| \Gamma | |
| j |
| i. \end{align} | |
| \Gamma | |
| λ |
This is a vertical-valued horizontal two-form on .
Given the connection and the soldering form, a torsion of with respect to is defined as
T=d\Gamma\sigma=\left(\partialλ\sigma
| i | |
| \mu |
+
| j\partial | |
| \Gamma | |
| j\sigma |
| i | |
| \mu |
-\partialj\Gamma
| j\right) | |
| \mu |
dxλ\wedgedx\mu ⊗ \partiali.
Let be a principal bundle with a structure Lie group . A principal connection on usually is described by a Lie algebra-valued connection one-form on . At the same time, a principal connection on is a global section of the jet bundle which is equivariant with respect to the canonical right action of in . Therefore, it is represented by a global section of the quotient bundle, called the bundle of principal connections. It is an affine bundle modelled on the vector bundle whose typical fiber is the Lie algebra of structure group, and where acts on by the adjoint representation. There is the canonical imbedding of to the quotient bundle which also is called the bundle of principal connections.
Given a basis for a Lie algebra of, the fiber bundle is endowed with bundle coordinates, and its sections are represented by vector-valued one-forms
A=dxλ ⊗ \left(\partialλ+
| m | |
| a | |
| λ |
{e}m\right),
where
| m | |
| a | |
| λ |
dxλ ⊗ {e}m
are the familiar local connection forms on .
Let us note that the jet bundle of is a configuration space of Yang–Mills gauge theory. It admits the canonical decomposition
\begin{align}
| r | |
| a | |
| λ\mu |
&=
| r | |
| \tfrac12\left(F | |
| λ\mu |
+
| r\right) | |
| S | |
| λ\mu |
\\ &=
| r | |
| \tfrac12\left(a | |
| λ\mu |
+
| r | |
| a | |
| \muλ |
-
| r | |
| c | |
| pq |
| p | |
| a | |
| λ |
| q\right) | |
| a | |
| \mu |
+
| r | |
| \tfrac12\left(a | |
| λ\mu |
-
| r | |
| a | |
| \muλ |
+
| r | |
| c | |
| pq |
| p | |
| a | |
| λ |
| q\right), | |
| a | |
| \mu |
\end{align}
where
F=\tfrac{1}{2}
| m | |
| F | |
| λ\mu |
dxλ\wedgedx\mu ⊗ {e}m
is called the strength form of a principal connection.