Editing Parallel transport (section)

===Metric connection===

A [[metric connection]] is any connection whose parallel transport mappings preserve the Riemannian metric, that is, for any curve <math>\gamma</math> and any two vectors <math>X, Y \in T_{\gamma(s)}M</math>,
:<math>\langle\Gamma(\gamma)_s^tX,\Gamma(\gamma)_s^tY\rangle_{\gamma(t)}=\langle X,Y\rangle_{\gamma(s)}.</math>

Taking the derivative at ''t'' = 0, the operator &nabla; satisfies a product rule with respect to the metric, namely
:<math>Z\langle X,Y\rangle = \langle \nabla_ZX,Y\rangle + \langle X,\nabla_Z Y\rangle.</math>

====Relationship to geodesics====

An affine connection distinguishes a class of curves called (affine) [[geodesic]]s.<ref>{{harv|Kobayashi|Nomizu|1996|loc=Volume 1, Chapter III}}</ref> A smooth curve <math>\gamma:I\rightarrow M</math> is an '''affine geodesic''' if <math>\dot\gamma</math> is parallel transported along <math>\gamma</math>, that is
:<math>\Gamma(\gamma)_s^t\dot\gamma(s) = \dot\gamma(t).\,</math>

Taking the derivative with respect to time, this takes the more familiar form
:<math>\nabla_{\dot\gamma(t)}\dot\gamma = 0.\,</math>

If <math>\nabla</math> is a metric connection, then the affine geodesics are the usual [[geodesic]]s of Riemannian geometry and are the locally distance minimizing curves. More precisely, first note that if <math>\gamma:I\rightarrow M</math>, where <math>I</math> is an open interval, is a geodesic, then the norm of <math>\dot\gamma</math> is constant on <math>I</math>. Indeed,
:<math>\frac{d}{dt}\langle\dot\gamma(t),\dot\gamma(t)\rangle = 2\langle\nabla_{\dot\gamma(t)}\dot\gamma(t),\dot\gamma(t)\rangle =0.</math>

It follows from an application of [[Gauss's lemma (Riemannian geometry)|Gauss's lemma]] that if <math>A</math> is the norm of <math>\dot\gamma(t)</math> then the distance, induced by the metric, between two ''close enough'' points on the curve <math>\gamma</math>, say <math>\gamma(t_1)</math> and <math>\gamma(t_2)</math>, is given by
<math display="block">\mbox{dist}\big(\gamma(t_1),\gamma(t_2)\big) = A|t_1 - t_2|.</math>
The formula above might not be true for points which are not close enough since the geodesic might for example wrap around the manifold (e.g. on a sphere).