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:

${\displaystyle 0\to \mathrm {V} Y\to \mathrm {T} Y\to Y\times _{X}\mathrm {T} X\to 0\,,}$

(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

${\displaystyle \Gamma :Y\times _{X}\mathrm {T} X\to \mathrm {T} Y}$

(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

${\displaystyle \mathrm {H} Y=\Gamma \left(Y\times _{X}\mathrm {T} X\right)\subset \mathrm {T} Y}$

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

${\displaystyle {\begin{aligned}\mathbb {R} \supset [,]\ni t&\to x(t)\in X\\\mathbb {R} \ni t&\to y(t)\in Y\end{aligned))}$

be two smooth paths in X and Y, respectively. Then t → y(t) is called the horizontal lift of x(t) if

${\displaystyle \pi (y(t))=x(t)\,,\qquad {\dot {y))(t)\in \mathrm {H} Y\,,\qquad t\in \mathbb {R} \,.}$

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, yⁱ), and let Γ be a connection on Y → X. It yields uniquely the horizontal tangent-valued one-form

${\displaystyle \Gamma =dx^{\mu }\otimes \left(\partial _{\mu }+\Gamma _{\mu }^{i}\left(x^{\nu },y^{j}\right)\partial _{i}\right)}$

(3)

on Y which projects onto the canonical tangent-valued form (tautological one-form or solder form)

${\displaystyle \theta _{X}=dx^{\mu }\otimes \partial _{\mu ))$

on X, and vice versa. With this form, the horizontal splitting 2 reads

${\displaystyle \Gamma :\partial _{\mu }\to \partial _{\mu }\rfloor \Gamma =\partial _{\mu }+\Gamma _{\mu }^{i}\partial _{i}\,.}$

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

${\displaystyle \Gamma \tau =\tau \rfloor \Gamma =\tau ^{\mu }\left(\partial _{\mu }+\Gamma _{\mu }^{i}\partial _{i}\right)\subset \mathrm {H} Y}$

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

${\displaystyle 0\to Y\times _{X}\mathrm {T} ^{*}X\to \mathrm {T} ^{*}Y\to \mathrm {V} ^{*}Y\to 0\,,}$

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

${\displaystyle \Gamma =\left(dy^{i}-\Gamma _{\lambda }^{i}dx^{\lambda }\right)\otimes \partial _{i}\,,}$

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

${\displaystyle f*\Gamma =\left(dy^{i}-\left(\Gamma \circ {\tilde {f))\right)_{\lambda }^{i}{\frac {\partial f^{\lambda )){\partial x'^{\mu ))}dx'^{\mu }\right)\otimes \partial _{i))$

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 (x^μ, yⁱ, yⁱ
_μ). Due to the canonical imbedding

${\displaystyle \mathrm {J} ^{1}Y\to _{Y}\left(Y\times _{X}\mathrm {T} ^{*}X\right)\otimes _{Y}\mathrm {T} Y\,,\qquad \left(y_{\mu }^{i}\right)\to dx^{\mu }\otimes \left(\partial _{\mu }+y_{\mu }^{i}\partial _{i}\right)\,,}$

any connection Γ 3 on a fibered manifold Y → X is represented by a global section

${\displaystyle \Gamma :Y\to \mathrm {J} ^{1}Y\,,\qquad y_{\lambda }^{i}\circ \Gamma =\Gamma _{\lambda }^{i}\,,}$

of the jet bundle J¹Y → Y, and vice versa. It is an affine bundle modelled on a vector bundle

${\displaystyle \left(Y\times _{X}T^{*}X\right)\otimes _{Y}\mathrm {V} Y\to Y\,.}$

(4)

There are the following corollaries of this fact.

Curvature and torsion

Given the connection Γ 3 on a fibered manifold Y → X, its curvature is defined as the Nijenhuis differential

${\displaystyle {\begin{aligned}R&={\tfrac {1}{2))d_{\Gamma }\Gamma \\&={\tfrac {1}{2))[\Gamma ,\Gamma ]_{\mathrm {FN} }\\&={\tfrac {1}{2))R_{\lambda \mu }^{i}\,dx^{\lambda }\wedge dx^{\mu }\otimes \partial _{i}\,,\\R_{\lambda \mu }^{i}&=\partial _{\lambda }\Gamma _{\mu }^{i}-\partial _{\mu }\Gamma _{\lambda }^{i}+\Gamma _{\lambda }^{j}\partial _{j}\Gamma _{\mu }^{i}-\Gamma _{\mu }^{j}\partial _{j}\Gamma _{\lambda }^{i}\,.\end{aligned))}$

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

${\displaystyle T=d_{\Gamma }\sigma =\left(\partial _{\lambda }\sigma _{\mu }^{i}+\Gamma _{\lambda }^{j}\partial _{j}\sigma _{\mu }^{i}-\partial _{j}\Gamma _{\lambda }^{i}\sigma _{\mu }^{j}\right)\,dx^{\lambda }\wedge dx^{\mu }\otimes \partial _{i}\,.}$

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 {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

${\displaystyle A=dx^{\lambda }\otimes \left(\partial _{\lambda }+a_{\lambda }^{m}{\mathrm {e} }_{m}\right)\,,}$

where

${\displaystyle a_{\lambda }^{m}\,dx^{\lambda }\otimes {\mathrm {e} }_{m))$

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

${\displaystyle {\begin{aligned}a_{\lambda \mu }^{r}&={\tfrac {1}{2))\left(F_{\lambda \mu }^{r}+S_{\lambda \mu }^{r}\right)\\&={\tfrac {1}{2))\left(a_{\lambda \mu }^{r}+a_{\mu \lambda }^{r}-c_{pq}^{r}a_{\lambda }^{p}a_{\mu }^{q}\right)+{\tfrac {1}{2))\left(a_{\lambda \mu }^{r}-a_{\mu \lambda }^{r}+c_{pq}^{r}a_{\lambda }^{p}a_{\mu }^{q}\right)\,,\end{aligned))}$

where

${\displaystyle F={\tfrac {1}{2))F_{\lambda \mu }^{m}\,dx^{\lambda }\wedge dx^{\mu }\otimes {\mathrm {e} }_{m))$

is called the strength form of a principal connection.