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 $Y$:

where $TY$ and $TX$ are the tangent bundles of $$, 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

over $Y$ which splits the exact sequence $$. 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 $Y$ on $$ onto $τ$, but need not defines the similar lift of a path in $X$ into $Y$. Let


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

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

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 $$ reads


 * $$\Gamma:\partial_\mu\to \partial_\mu\rfloor\Gamma=\partial_\mu +\Gamma^i_\mu\partial_i\,. $$

In particular, the connection $Γ$ in $Y$ yields the horizontal lift of any vector field $τ = τ^{μ} ∂_{μ}$ on $X$ to a projectable vector field


 * $$\Gamma \tau=\tau\rfloor\Gamma=\tau^\mu\left(\partial_\mu +\Gamma^i_\mu\partial_i\right)\subset \mathrm{H}Y$$

on $$.

Connection as a vertical-valued form
The horizontal splitting $$ of the exact sequence $X$ 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


 * $$\Gamma= \left(dy^i -\Gamma^i_\lambda dx^\lambda\right)\otimes\partial_i\,, $$

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 $$ by $$. Then any connection $Γ$ $Y$ on $Y → X$ induces the pullback connection


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

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 $(x^{μ}, y^{i}, yi μ)$. Due to the canonical imbedding


 * $$ \mathrm{J}^1Y\to_Y \left(Y\times_X \mathrm{T}^*X \right)\otimes_Y \mathrm{T}Y\,, \qquad \left(y^i_\mu\right)\to dx^\mu\otimes \left(\partial_\mu + y^i_\mu\partial_i\right)\,, $$

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


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

of the jet bundle $J^{1}Y → Y$, and vice versa. It is an affine bundle modelled on a vector bundle

There are the following corollaries of this fact.

1. Connections on a fibered manifold $Y → X$ make up an affine space modelled on the vector space of soldering forms

on $Y → X$, i.e., sections of the vector bundle $f$.

2. Connection coefficients possess the coordinate transformation law


 * ${\Gamma'}^i_\lambda = \frac{\partial x^\mu}{\partial {x'}^\lambda}\left(\partial_\mu {y'}^i+\Gamma^j_\mu\partial_j{y'}^i\right)\,. $

3. Every connection $Γ$ on a fibred manifold $Y → X$ yields the first order differential operator


 * $D_\Gamma:\mathrm{J}^1Y\to_Y \mathrm{T}^*X\otimes_Y \mathrm{V}Y\,, \qquad D_\Gamma = \left(y^i_\lambda -\Gamma^i_\lambda\right)dx^\lambda\otimes\partial_i\,, $

on $$ called the covariant differential relative to the connection $Γ$. If $s : X → Y$ is a section, its covariant differential


 * $\nabla^\Gamma s = \left(\partial_\lambda s^i - \Gamma_\lambda^i\circ s\right) dx^\lambda\otimes \partial_i\,, $

and the covariant derivative


 * $\nabla_\tau^\Gamma s=\tau\rfloor\nabla^\Gamma s$

along a vector field $$ on $$ are defined.

Curvature and torsion
Given the connection $Γ$ $$ on a fibered manifold $Y → X$, its curvature is defined as the Nijenhuis differential


 * $$\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}$$

This is a vertical-valued horizontal two-form on $$.

Given the connection $Γ$ $Y$ and the soldering form $τ$ $X$, a torsion of $Γ$ with respect to $$ 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 $Y$. A principal connection on $$ usually is described by a Lie algebra-valued connection one-form on $σ$. At the same time, a principal connection on $$ is a global section of the jet bundle $J^{1}P → P$ which is equivariant with respect to the canonical right action of $σ$ in $G$. 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 $P$, and where $P$ acts on by the adjoint representation. There is the canonical imbedding of $P$ to the quotient bundle $TP/G$ which also is called the bundle of principal connections.

Given a basis ${e_{m}}|undefined$ for a Lie algebra of $G$, the fiber bundle $P$ is endowed with bundle coordinates $(x^{μ}, am μ)$, and its sections are represented by vector-valued one-forms


 * $$A=dx^\lambda\otimes \left(\partial_\lambda + a^m_\lambda {\mathrm e}_m\right)\,, $$

where
 * $$ a^m_\lambda \, dx^\lambda\otimes {\mathrm e}_m $$

are the familiar local connection forms on $G$.

Let us note that the jet bundle $J^{1}C$ of $G$ is a configuration space of Yang–Mills gauge theory. It admits the canonical decomposition


 * $$\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}$$

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.