Английская Википедия:Connection (fibred manifold)
Шаблон:Short description Шаблон:Technical
In differential geometry, a fibered manifold is surjective submersion of smooth manifolds Шаблон:Math. 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.
Formal definition
Let Шаблон:Math be a fibered manifold. A generalized connection on Шаблон:Mvar is a section Шаблон:Math, where Шаблон:Math is the jet manifold of Шаблон:Mvar.[1]
Connection as a horizontal splitting
With the above manifold Шаблон:Mvar there is the following canonical short exact sequence of vector bundles over Шаблон:Mvar:
where Шаблон:Math and Шаблон:Math are the tangent bundles of Шаблон:Mvar, respectively, Шаблон:Math is the vertical tangent bundle of Шаблон:Mvar, and Шаблон:Math is the pullback bundle of Шаблон:Math onto Шаблон:Mvar.
A connection on a fibered manifold Шаблон:Math is defined as a linear bundle morphism
over Шаблон:Mvar which splits the exact sequence Шаблон:EquationRef. A connection always exists.
Sometimes, this connection Шаблон:Math is called the Ehresmann connection because it yields the horizontal distribution
- <math>\mathrm{H}Y=\Gamma\left(Y\times_X \mathrm{T}X \right) \subset \mathrm{T}Y</math>
of Шаблон:Math and its horizontal decomposition Шаблон:Math.
At the same time, by an Ehresmann connection also is meant the following construction. Any connection Шаблон:Math on a fibered manifold Шаблон:Math yields a horizontal lift Шаблон:Math of a vector field Шаблон:Mvar on Шаблон:Mvar onto Шаблон:Mvar, but need not defines the similar lift of a path in Шаблон:Mvar into Шаблон:Mvar. Let
- <math>\begin{align}\mathbb R\supset[,]\ni t&\to x(t)\in X \\ \mathbb R\ni t&\to y(t)\in Y\end{align}</math>
be two smooth paths in Шаблон:Mvar and Шаблон:Mvar, respectively. Then Шаблон:Math is called the horizontal lift of Шаблон:Math if
- <math>\pi(y(t))= x(t)\,, \qquad \dot y(t)\in \mathrm{H}Y \,, \qquad t\in\mathbb R\,.</math>
A connection Шаблон:Math is said to be the Ehresmann connection if, for each path Шаблон:Math in Шаблон:Mvar, there exists its horizontal lift through any point Шаблон:Math. A fibered manifold is a fiber bundle if and only if it admits such an Ehresmann connection.
Connection as a tangent-valued form
Given a fibered manifold Шаблон:Math, let it be endowed with an atlas of fibered coordinates Шаблон:Math, and let Шаблон:Math be a connection on Шаблон:Math. It yields uniquely the horizontal tangent-valued one-form
on Шаблон:Mvar which projects onto the canonical tangent-valued form (tautological one-form or solder form)
- <math>\theta_X=dx^\mu\otimes\partial_\mu</math>
on Шаблон:Mvar, and vice versa. With this form, the horizontal splitting Шаблон:EquationNote reads
- <math>\Gamma:\partial_\mu\to \partial_\mu\rfloor\Gamma=\partial_\mu +\Gamma^i_\mu\partial_i\,. </math>
In particular, the connection Шаблон:Math in Шаблон:EquationNote yields the horizontal lift of any vector field Шаблон:Math on Шаблон:Mvar to a projectable vector field
- <math>\Gamma \tau=\tau\rfloor\Gamma=\tau^\mu\left(\partial_\mu +\Gamma^i_\mu\partial_i\right)\subset \mathrm{H}Y</math>
on Шаблон:Mvar.
Connection as a vertical-valued form
The horizontal splitting Шаблон:EquationNote of the exact sequence Шаблон:EquationNote defines the corresponding splitting of the dual exact sequence
- <math>0\to Y\times_X \mathrm{T}^*X \to \mathrm{T}^*Y\to \mathrm{V}^*Y\to 0\,,</math>
where Шаблон:Math and Шаблон:Math are the cotangent bundles of Шаблон:Mvar, respectively, and Шаблон:Math is the dual bundle to Шаблон:Math, called the vertical cotangent bundle. This splitting is given by the vertical-valued form
- <math>\Gamma= \left(dy^i -\Gamma^i_\lambda dx^\lambda\right)\otimes\partial_i\,, </math>
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 Шаблон:Math, let Шаблон:Math be a morphism and Шаблон:Math the pullback bundle of Шаблон:Mvar by Шаблон:Mvar. Then any connection Шаблон:Math Шаблон:EquationNote on Шаблон:Math induces the pullback connection
- <math>f*\Gamma=\left(dy^i-\left(\Gamma\circ \tilde f\right)^i_\lambda\frac{\partial f^\lambda}{\partial x'^\mu}dx'^\mu\right)\otimes\partial_i </math>
on Шаблон:Math.
Connection as a jet bundle section
Let Шаблон:Math be the jet manifold of sections of a fibered manifold Шаблон:Math, with coordinates Шаблон:Math. Due to the canonical imbedding
- <math> \mathrm{J}^1Y\to_Y \left(Y\times_X \mathrm{T}^*X \right)\otimes_Y \mathrm{T}Y\,, \qquad \left(y^i_\mu\right)\to dx^\mu\otimes \left(\partial_\mu + y^i_\mu\partial_i\right)\,, </math>
any connection Шаблон:Math Шаблон:EquationNote on a fibered manifold Шаблон:Math is represented by a global section
- <math>\Gamma :Y\to \mathrm{J}^1Y\,, \qquad y_\lambda^i\circ\Gamma=\Gamma_\lambda^i\,, </math>
of the jet bundle Шаблон:Math, and vice versa. It is an affine bundle modelled on a vector bundle
There are the following corollaries of this fact.
Curvature and torsion
Given the connection Шаблон:Math Шаблон:EquationNote on a fibered manifold Шаблон:Math, its curvature is defined as the Nijenhuis differential
- <math>\begin{align}
R&=\tfrac12 d_\Gamma\Gamma\\&=\tfrac12 [\Gamma,\Gamma]_\mathrm{FN} \\&= \tfrac12 R_{\lambda\mu}^i \, dx^\lambda\wedge dx^\mu\otimes\partial_i\,, \\ R_{\lambda\mu}^i &= \partial_\lambda\Gamma_\mu^i - \partial_\mu\Gamma_\lambda^i + \Gamma_\lambda^j\partial_j \Gamma_\mu^i - \Gamma_\mu^j\partial_j \Gamma_\lambda^i\,. \end{align}</math>
This is a vertical-valued horizontal two-form on Шаблон:Mvar.
Given the connection Шаблон:Math Шаблон:EquationNote and the soldering form Шаблон:Mvar Шаблон:EquationNote, a torsion of Шаблон:Math with respect to Шаблон:Mvar is defined as
- <math>T = d_\Gamma \sigma = \left(\partial_\lambda\sigma_\mu^i + \Gamma_\lambda^j\partial_j\sigma_\mu^i -\partial_j\Gamma_\lambda^i\sigma_\mu^j\right) \, dx^\lambda\wedge dx^\mu\otimes \partial_i\,. </math>
Bundle of principal connections
Let Шаблон:Math be a principal bundle with a structure Lie group Шаблон:Mvar. A principal connection on Шаблон:Mvar usually is described by a Lie algebra-valued connection one-form on Шаблон:Mvar. At the same time, a principal connection on Шаблон:Mvar is a global section of the jet bundle Шаблон:Math which is equivariant with respect to the canonical right action of Шаблон:Mvar in Шаблон:Mvar. Therefore, it is represented by a global section of the quotient bundle Шаблон:Math, called the bundle of principal connections. It is an affine bundle modelled on the vector bundle Шаблон:Math whose typical fiber is the Lie algebra Шаблон:Math of structure group Шаблон:Mvar, and where Шаблон:Mvar acts on by the adjoint representation. There is the canonical imbedding of Шаблон:Mvar to the quotient bundle Шаблон:Math which also is called the bundle of principal connections.
Given a basis Шаблон:Math} for a Lie algebra of Шаблон:Mvar, the fiber bundle Шаблон:Mvar is endowed with bundle coordinates Шаблон:Math, and its sections are represented by vector-valued one-forms
- <math>A=dx^\lambda\otimes \left(\partial_\lambda + a^m_\lambda {\mathrm e}_m\right)\,, </math>
where
- <math> a^m_\lambda \, dx^\lambda\otimes {\mathrm e}_m </math>
are the familiar local connection forms on Шаблон:Mvar.
Let us note that the jet bundle Шаблон:Math of Шаблон:Mvar is a configuration space of Yang–Mills gauge theory. It admits the canonical decomposition
- <math>\begin{align} a_{\lambda\mu}^r &= \tfrac12\left(F_{\lambda\mu}^r + S_{\lambda\mu}^r\right) \\
&= \tfrac12\left(a_{\lambda\mu}^r + a_{\mu\lambda}^r - c_{pq}^r a_\lambda^p a_\mu^q\right) + \tfrac12\left(a_{\lambda\mu}^r - a_{\mu\lambda}^r + c_{pq}^r a_\lambda^p a_\mu^q\right)\,, \end{align}</math>
where
- <math> F=\tfrac{1}{2} F_{\lambda\mu}^m \, dx^\lambda\wedge dx^\mu\otimes {\mathrm e}_m </math>
is called the strength form of a principal connection.
See also
Notes
References
