Operation on fibered manifolds
In differential geometry, a fibered manifold is surjective submersion of smooth manifolds Y → X. 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 π : Y → X be a fibered manifold. A generalized connection on Y is a section Γ : Y → J1Y, where J1Y is the jet manifold of Y.[1]
Connection as a horizontal splitting
With the above manifold π there is the following canonical short exact sequence of vector bundles over Y:
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![{\displaystyle 0\to \mathrm {V} Y\to \mathrm {T} Y\to Y\times _{X}\mathrm {T} X\to 0\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf16a6aef146a52ee4458f03390709d930a3c91)
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(1)
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where TY and TX are the tangent bundles of Y, respectively, VY is the vertical tangent bundle of Y, and Y ×X TX is the pullback bundle of TX onto Y.
A connection on a fibered manifold Y → X is defined as a linear bundle morphism
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![{\displaystyle \Gamma :Y\times _{X}\mathrm {T} X\to \mathrm {T} Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d73b11e08df23b456ee9491df7aa4c4681f39c6b)
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(2)
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over Y which splits the exact sequence 1. A connection always exists.
Sometimes, this connection Γ is called the Ehresmann connection because it yields the horizontal distribution
![{\displaystyle \mathrm {H} Y=\Gamma \left(Y\times _{X}\mathrm {T} X\right)\subset \mathrm {T} Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d56b7b96364d1ecdde0122c2bb2b16ba3036505e)
of TY and its horizontal decomposition TY = VY ⊕ HY.
At the same time, by an Ehresmann connection also is meant the following construction. Any connection Γ on a fibered manifold Y → X yields a horizontal lift Γ ∘ τ of a vector field τ on X onto Y, but need not defines the similar lift of a path in X into Y. Let
![{\displaystyle {\begin{aligned}\mathbb {R} \supset [,]\ni t&\to x(t)\in X\\\mathbb {R} \ni t&\to y(t)\in Y\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d568dce94ede7fb9805eebfc7eb840a2c5ce759)
be two smooth paths in X and Y, respectively. Then t → y(t) is called the horizontal lift of x(t) if
![{\displaystyle \pi (y(t))=x(t)\,,\qquad {\dot {y))(t)\in \mathrm {H} Y\,,\qquad t\in \mathbb {R} \,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9b3076f1e496192970ef9f1001f47a38e34a6f)
A connection Γ is said to be the Ehresmann connection if, for each path x([0,1]) in X, there exists its horizontal lift through any point y ∈ π−1(x([0,1])). 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 Y → X, let it be endowed with an atlas of fibered coordinates (xμ, yi), and let Γ be a connection on Y → X. It yields uniquely the horizontal tangent-valued one-form
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![{\displaystyle \Gamma =dx^{\mu }\otimes \left(\partial _{\mu }+\Gamma _{\mu }^{i}\left(x^{\nu },y^{j}\right)\partial _{i}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb90bfe454a769c11c20d941dbac1e87e2da859b)
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(3)
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on Y which projects onto the canonical tangent-valued form (tautological one-form or solder form)
![{\displaystyle \theta _{X}=dx^{\mu }\otimes \partial _{\mu ))](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7004f2fc457235e38c52b7cc2b8a7d55661acd)
on X, and vice versa. With this form, the horizontal splitting 2 reads
![{\displaystyle \Gamma :\partial _{\mu }\to \partial _{\mu }\rfloor \Gamma =\partial _{\mu }+\Gamma _{\mu }^{i}\partial _{i}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c465ba3ff326952f3a851b17b7fca28e0e6d2e6)
In particular, the connection Γ in 3 yields the horizontal lift of any vector field τ = τμ ∂μ on X to a projectable vector field
![{\displaystyle \Gamma \tau =\tau \rfloor \Gamma =\tau ^{\mu }\left(\partial _{\mu }+\Gamma _{\mu }^{i}\partial _{i}\right)\subset \mathrm {H} Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7de487aa4be12a5a861a3096445b232d320277d)
on Y.
Connection as a vertical-valued form
The horizontal splitting 2 of the exact sequence 1 defines the corresponding splitting of the dual exact sequence
![{\displaystyle 0\to Y\times _{X}\mathrm {T} ^{*}X\to \mathrm {T} ^{*}Y\to \mathrm {V} ^{*}Y\to 0\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53906112059f6efbde2835a365d1c212b2cb385c)
where T*Y and T*X are the cotangent bundles of Y, respectively, and V*Y → Y is the dual bundle to VY → Y, called the vertical cotangent bundle. This splitting is given by the vertical-valued form
![{\displaystyle \Gamma =\left(dy^{i}-\Gamma _{\lambda }^{i}dx^{\lambda }\right)\otimes \partial _{i}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14368e3add410109da724c0a1ef33806c789ed7b)
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 Y → X, let f : X′ → X be a morphism and f ∗ Y → X′ the pullback bundle of Y by f. Then any connection Γ 3 on Y → X induces the pullback connection
![{\displaystyle f*\Gamma =\left(dy^{i}-\left(\Gamma \circ {\tilde {f))\right)_{\lambda }^{i}{\frac {\partial f^{\lambda )){\partial x'^{\mu ))}dx'^{\mu }\right)\otimes \partial _{i))](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8852ea7f0b10125f0dc8e9661702826d8beb2e)
on f ∗ Y → X′.
Connection as a jet bundle section
Let J1Y be the jet manifold of sections of a fibered manifold Y → X, with coordinates (xμ, yi, yi
μ). Due to the canonical imbedding
![{\displaystyle \mathrm {J} ^{1}Y\to _{Y}\left(Y\times _{X}\mathrm {T} ^{*}X\right)\otimes _{Y}\mathrm {T} Y\,,\qquad \left(y_{\mu }^{i}\right)\to dx^{\mu }\otimes \left(\partial _{\mu }+y_{\mu }^{i}\partial _{i}\right)\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca74372ad76caa1b05b72ff4dae90d54e807019)
any connection Γ 3 on a fibered manifold Y → X is represented by a global section
![{\displaystyle \Gamma :Y\to \mathrm {J} ^{1}Y\,,\qquad y_{\lambda }^{i}\circ \Gamma =\Gamma _{\lambda }^{i}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df2a004386d70cdbe1964934b1642b0a6f95f810)
of the jet bundle J1Y → Y, and vice versa. It is an affine bundle modelled on a vector bundle
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![{\displaystyle \left(Y\times _{X}T^{*}X\right)\otimes _{Y}\mathrm {V} Y\to Y\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1082ef56c0a2da5c309939039d9deb21420dfba)
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(4)
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There are the following corollaries of this fact.
- Connections on a fibered manifold Y → X make up an affine space modelled on the vector space of soldering forms
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![{\displaystyle \sigma =\sigma _{\mu }^{i}dx^{\mu }\otimes \partial _{i))](https://wikimedia.org/api/rest_v1/media/math/render/svg/a62d561cc1ca40b63bc8caea0e1d3d011accffa6)
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(5)
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on Y → X, i.e., sections of the vector bundle 4. - Connection coefficients possess the coordinate transformation law
![{\displaystyle {\Gamma '}_{\lambda }^{i}={\frac {\partial x^{\mu )){\partial {x'}^{\lambda ))}\left(\partial _{\mu }{y'}^{i}+\Gamma _{\mu }^{j}\partial _{j}{y'}^{i}\right)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d78bf0e7da28e4db111918b4afb0639336ba9ab4)
- Every connection Γ on a fibred manifold Y → X yields the first order differential operator
![{\displaystyle D_{\Gamma }:\mathrm {J} ^{1}Y\to _{Y}\mathrm {T} ^{*}X\otimes _{Y}\mathrm {V} Y\,,\qquad D_{\Gamma }=\left(y_{\lambda }^{i}-\Gamma _{\lambda }^{i}\right)dx^{\lambda }\otimes \partial _{i}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d6d2dee97384a72f7ce49417f5296932ba5eb5a)
on Y called the covariant differential relative to the connection Γ. If s : X → Y is a section, its covariant differential
![{\displaystyle \nabla ^{\Gamma }s=\left(\partial _{\lambda }s^{i}-\Gamma _{\lambda }^{i}\circ s\right)dx^{\lambda }\otimes \partial _{i}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5baf01a42c73a08918be0119365410c399e0bf58)
and the covariant derivative
![{\displaystyle \nabla _{\tau }^{\Gamma }s=\tau \rfloor \nabla ^{\Gamma }s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec1cae702f3af49a60dc94e753b2353539c3365)
along a vector field τ on X are defined.
Curvature and torsion
Given the connection Γ 3 on a fibered manifold Y → X, its curvature is defined as the Nijenhuis differential
![{\displaystyle {\begin{aligned}R&={\tfrac {1}{2))d_{\Gamma }\Gamma \\&={\tfrac {1}{2))[\Gamma ,\Gamma ]_{\mathrm {FN} }\\&={\tfrac {1}{2))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{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba4042d4b3f07bcc222f8bd571d79a2edcdebaa)
This is a vertical-valued horizontal two-form on Y.
Given the connection Γ 3 and the soldering form σ 5, a torsion of Γ with respect to σ is defined as
![{\displaystyle 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}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2733412e34b5d2f5d1163bebf7ea0536890ce4dc)
Bundle of principal connections
Let π : P → M be a principal bundle with a structure Lie group G. A principal connection on P usually is described by a Lie algebra-valued connection one-form on P. At the same time, a principal connection on P is a global section of the jet bundle J1P → P which is equivariant with respect to the canonical right action of G in P. Therefore, it is represented by a global section of the quotient bundle C = J1P/G → M, called the bundle of principal connections. It is an affine bundle modelled on the vector bundle VP/G → M whose typical fiber is the Lie algebra g of structure group G, and where G acts on by the adjoint representation. There is the canonical imbedding of C to the quotient bundle TP/G which also is called the bundle of principal connections.
Given a basis {em} for a Lie algebra of G, the fiber bundle C is endowed with bundle coordinates (xμ, am
μ), and its sections are represented by vector-valued one-forms
![{\displaystyle A=dx^{\lambda }\otimes \left(\partial _{\lambda }+a_{\lambda }^{m}{\mathrm {e} }_{m}\right)\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/653e17bcff8530f276139a9b07804e5ff40aaae0)
where
![{\displaystyle a_{\lambda }^{m}\,dx^{\lambda }\otimes {\mathrm {e} }_{m))](https://wikimedia.org/api/rest_v1/media/math/render/svg/088c7030eaa64976ccbdc76ea796e4546a205723)
are the familiar local connection forms on M.
Let us note that the jet bundle J1C of C is a configuration space of Yang–Mills gauge theory. It admits the canonical decomposition
![{\displaystyle {\begin{aligned}a_{\lambda \mu }^{r}&={\tfrac {1}{2))\left(F_{\lambda \mu }^{r}+S_{\lambda \mu }^{r}\right)\\&={\tfrac {1}{2))\left(a_{\lambda \mu }^{r}+a_{\mu \lambda }^{r}-c_{pq}^{r}a_{\lambda }^{p}a_{\mu }^{q}\right)+{\tfrac {1}{2))\left(a_{\lambda \mu }^{r}-a_{\mu \lambda }^{r}+c_{pq}^{r}a_{\lambda }^{p}a_{\mu }^{q}\right)\,,\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bad864ff648ad19816dbe6f7bf38936051722404)
where
![{\displaystyle F={\tfrac {1}{2))F_{\lambda \mu }^{m}\,dx^{\lambda }\wedge dx^{\mu }\otimes {\mathrm {e} }_{m))](https://wikimedia.org/api/rest_v1/media/math/render/svg/99336726457ada601d520229c9366921679e4c89)
is called the strength form of a principal connection.