Editing Connection (vector bundle) (section)

==Parallel transport and holonomy==

A connection <math>\nabla</math> on a vector bundle <math>E\to M</math> defines a notion of [[parallel transport]] on <math>E</math> along a curve in <math>M</math>. Let <math>\gamma: [0,1]\to M</math> be a smooth [[path (topology)|path]] in <math>M</math>. A section <math>s</math> of <math>E</math> along <math>\gamma</math> is said to be '''parallel''' if
:<math>\nabla_{\dot\gamma(t)}s = 0</math>
for all <math>t\in [0,1]</math>. Equivalently, one can consider the [[pullback bundle]] <math>\gamma^* E</math> of <math>E</math> by <math>\gamma</math>. This is a vector bundle over <math>[0,1]</math> with fiber <math>E_{\gamma(t)}</math> over <math>t\in [0,1]</math>. The connection <math>\nabla</math> on <math>E</math> pulls back to a connection on <math>\gamma^* E</math>. A section <math>s</math> of  <math>\gamma^* E</math> is parallel if and only if <math>\gamma^* \nabla(s) = 0</math>.

Suppose <math>\gamma</math> is a path from <math>x</math> to <math>y</math> in <math>M</math>. The above equation defining parallel sections is a first-order [[ordinary differential equation]] (cf. [[#Local_expression|local expression]] above) and so has a unique solution for each possible initial condition. That is, for each vector <math>v</math> in <math>E_x</math> there exists a unique parallel section <math>s</math> of <math>\gamma^* E</math> with <math>s(0) = v</math>. Define a '''parallel transport map'''
:<math>\tau_\gamma : E_x \to E_y\,</math>
by <math>\tau_\gamma(v) = s(1)</math>. It can be shown that <math>\tau_\gamma</math> is a [[linear isomorphism]], with inverse given by following the same procedure with the reversed path <math>\gamma^-</math> from <math>y</math> to <math>x</math>.

[[File:Paralleltransportcovariantderivative.png|400px|thumb|right|How to recover the covariant derivative of a connection from its parallel transport. The values <math>s(\gamma(t))</math> of a section <math>s\in \Gamma(E)</math> are parallel transported along the path <math>\gamma</math> back to <math>\gamma(0)=x</math>, and then the covariant derivative is taken in the fixed vector space, the fibre <math>E_x</math> over <math>x</math>.]]

Parallel transport can be used to define the [[holonomy group]] of the connection <math>\nabla</math> based at a point <math>x</math> in <math>M</math>. This is the subgroup of <math>\operatorname{GL}(E_x)</math> consisting of all parallel transport maps coming from [[loop (topology)|loop]]s based at <math>x</math>:
:<math>\mathrm{Hol}_x = \{\tau_\gamma : \gamma \text{ is a loop based at } x\}.\,</math>
The holonomy group of a connection is intimately related to the curvature of the connection {{harv|AmbroseSinger|1953}}.

The connection can be recovered from its parallel transport operators as follows. If <math>X\in \Gamma(TM)</math> is a vector field and <math>s\in \Gamma(E)</math> a section, at a point <math>x\in M</math> pick an [[integral curve]] <math>\gamma: (-\varepsilon, \varepsilon) \to M</math> for <math>X</math> at <math>x</math>. For each <math>t\in (-\varepsilon, \varepsilon)</math> we will write <math>\tau_t : E_{\gamma(t)} \to E_x</math> for the parallel transport map traveling along <math>\gamma</math> from <math>t</math> to <math>0</math>. In particular for every <math>t\in (-\varepsilon, \varepsilon)</math>, we have <math>\tau_t s(\gamma(t)) \in E_x</math>. Then <math>t\mapsto \tau_t s(\gamma(t))</math> defines a curve in the vector space <math>E_x</math>, which may be differentiated. The covariant derivative is recovered as
:<math>\nabla_X s(x) = \frac{d}{dt} \left( \tau_t s(\gamma(t)) \right)_{t=0}.</math>
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms <math>\tau_{\gamma}</math> between fibres of <math>E</math> and taking the above expression as the definition of <math>\nabla</math>.