# Connection (fibred manifold)

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 → J$^1$Y, where J$^1$Y is the jet manifold of Y.

### Connection as a horizontal splitting

With the above manifold π there is the following canonical short exact sequence of vector bundles over Y:
| $0\to \mathrm {V} Y\to \mathrm {T} Y\to Y\times _{X}\mathrm {T} X\to 0\,,$ | (1) |
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
| $\Gamma :Y\times _{X}\mathrm {T} X\to \mathrm {T} Y$ | (2) |
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
    $\mathrm {H} Y=\Gamma \left(Y\times _{X}\mathrm {T} X\right)\subset \mathrm {T} Y$
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
    ${\begin{aligned}\mathbb {R} \supset [,]\ni t&\to x(t)\in X\\\mathbb {R} \ni t&\to y(t)\in Y\end{aligned}}$
be two smooth paths in X and Y, respectively. Then t → y(t) is called the horizontal lift of x(t) if
    $\pi (y(t))=x(t)\,,\qquad {\dot {y}}(t)\in \mathrm {H} Y\,,\qquad t\in \mathbb {R} \,.$
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 (x$^μ$, y$^i$), and let Γ be a connection on Y → X. It yields uniquely the horizontal tangent-valued one-form
| $\Gamma =dx^{\mu }\otimes \left(\partial _{\mu }+\Gamma _{\mu }^{i}\left(x^{\nu },y^{j}\right)\partial _{i}\right)$ | (3) |
on Y which projects onto the canonical tangent-valued form (tautological one-form or solder form)
    $\theta _{X}=dx^{\mu }\otimes \partial _{\mu }$
on X, and vice versa. With this form, the horizontal splitting 2 reads
    $\Gamma :\partial _{\mu }\to \partial _{\mu }\rfloor \Gamma =\partial _{\mu }+\Gamma _{\mu }^{i}\partial _{i}\,.$
In particular, the connection Γ in 3 yields the horizontal lift of any vector field τ = τ$^μ$ ∂$_μ$ on X to a projectable vector field
    $\Gamma \tau =\tau \rfloor \Gamma =\tau ^{\mu }\left(\partial _{\mu }+\Gamma _{\mu }^{i}\partial _{i}\right)\subset \mathrm {H} Y$
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
    $0\to Y\times _{X}\mathrm {T} ^{*}X\to \mathrm {T} ^{*}Y\to \mathrm {V} ^{*}Y\to 0\,,$
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
    $\Gamma =\left(dy^{i}-\Gamma _{\lambda }^{i}dx^{\lambda }\right)\otimes \partial _{i}\,,$
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 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
    $f*\Gamma =\left(dy^{i}-\left(\Gamma \circ {\tilde {f}}\right)_{\lambda }^{i}{\frac {\partial f^{\lambda }}{\partial x'^{\mu }}}dx'^{\mu }\right)\otimes \partial _{i}$
on f ∗ Y → X′.

### Connection as a jet bundle section

Let J$^1$Y be the jet manifold of sections of a fibered manifold Y → X, with coordinates (x$^μ$, y$^i$, y$^i$ $_μ$). Due to the canonical imbedding
    $\mathrm {J} ^{1}Y\to _{Y}\left(Y\times _{X}\mathrm {T} ^{*}X\right)\otimes _{Y}\mathrm {T} Y\,,\qquad \left(y_{\mu }^{i}\right)\to dx^{\mu }\otimes \left(\partial _{\mu }+y_{\mu }^{i}\partial _{i}\right)\,,$
any connection Γ 3 on a fibered manifold Y → X is represented by a global section
    $\Gamma :Y\to \mathrm {J} ^{1}Y\,,\qquad y_{\lambda }^{i}\circ \Gamma =\Gamma _{\lambda }^{i}\,,$
of the jet bundle J$^1$Y → Y, and vice versa. It is an affine bundle modelled on a vector bundle
| $\left(Y\times _{X}T^{*}X\right)\otimes _{Y}\mathrm {V} Y\to Y\,.$ | (4) |
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 $\sigma =\sigma _{\mu }^{i}dx^{\mu }\otimes \partial _{i}$ (5) on Y → X, i.e., sections of the vector bundle 4.
* Connection coefficients possess the coordinate transformation law
    ${\Gamma '}_{\lambda }^{i}={\frac {\partial x^{\mu }}{\partial {x'}^{\lambda }}}\left(\partial _{\mu }{y'}^{i}+\Gamma _{\mu }^{j}\partial _{j}{y'}^{i}\right)\,.$
* Every connection Γ on a fibred manifold Y → X yields the first order differential operator on Y called the covariant differential relative to the connection Γ. If s : X → Y is a section, its covariant differential and the covariant derivative along a vector field τ on X are defined.
    $D_{\Gamma }:\mathrm {J} ^{1}Y\to _{Y}\mathrm {T} ^{*}X\otimes _{Y}\mathrm {V} Y\,,\qquad D_{\Gamma }=\left(y_{\lambda }^{i}-\Gamma _{\lambda }^{i}\right)dx^{\lambda }\otimes \partial _{i}\,,$
    $\nabla ^{\Gamma }s=\left(\partial _{\lambda }s^{i}-\Gamma _{\lambda }^{i}\circ s\right)dx^{\lambda }\otimes \partial _{i}\,,$
    $\nabla _{\tau }^{\Gamma }s=\tau \rfloor \nabla ^{\Gamma }s$

## Curvature and torsion

Given the connection Γ 3 on a fibered manifold Y → X, its curvature is defined as the Nijenhuis differential
    ${\begin{aligned}R&={\tfrac {1}{2}}d_{\Gamma }\Gamma \\&={\tfrac {1}{2}}[\Gamma ,\Gamma ]_{\mathrm {FN} }\\&={\tfrac {1}{2}}R_{\lambda \mu }^{i}\,dx^{\lambda }\wedge dx^{\mu }\otimes \partial _{i}\,,\\R_{\lambda \mu }^{i}&=\partial _{\lambda }\Gamma _{\mu }^{i}-\partial _{\mu }\Gamma _{\lambda }^{i}+\Gamma _{\lambda }^{j}\partial _{j}\Gamma _{\mu }^{i}-\Gamma _{\mu }^{j}\partial _{j}\Gamma _{\lambda }^{i}\,.\end{aligned}}$
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
    $T=d_{\Gamma }\sigma =\left(\partial _{\lambda }\sigma _{\mu }^{i}+\Gamma _{\lambda }^{j}\partial _{j}\sigma _{\mu }^{i}-\partial _{j}\Gamma _{\lambda }^{i}\sigma _{\mu }^{j}\right)\,dx^{\lambda }\wedge dx^{\mu }\otimes \partial _{i}\,.$

## 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 J$^1$P → 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 = J$^1$P/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 {e$_m$} for a Lie algebra of G, the fiber bundle C is endowed with bundle coordinates (x$^μ$, a$^m$ $_μ$), and its sections are represented by vector-valued one-forms
    $A=dx^{\lambda }\otimes \left(\partial _{\lambda }+a_{\lambda }^{m}{\mathrm {e} }_{m}\right)\,,$
where
    $a_{\lambda }^{m}\,dx^{\lambda }\otimes {\mathrm {e} }_{m}$
are the familiar local connection forms on M.
Let us note that the jet bundle J$^1$C of C is a configuration space of Yang–Mills gauge theory. It admits the canonical decomposition
    ${\begin{aligned}a_{\lambda \mu }^{r}&={\tfrac {1}{2}}\left(F_{\lambda \mu }^{r}+S_{\lambda \mu }^{r}\right)\\&={\tfrac {1}{2}}\left(a_{\lambda \mu }^{r}+a_{\mu \lambda }^{r}-c_{pq}^{r}a_{\lambda }^{p}a_{\mu }^{q}\right)+{\tfrac {1}{2}}\left(a_{\lambda \mu }^{r}-a_{\mu \lambda }^{r}+c_{pq}^{r}a_{\lambda }^{p}a_{\mu }^{q}\right)\,,\end{aligned}}$
where
    $F={\tfrac {1}{2}}F_{\lambda \mu }^{m}\,dx^{\lambda }\wedge dx^{\mu }\otimes {\mathrm {e} }_{m}$
is called the strength form of a principal connection.