Editing Parallel transport (section)

==Parallel transport of tangent vectors==

Let <math>M</math> be a [[smooth manifold]]. For each point <math>p \in M</math>, there is an associated vector space <math>T_pM</math> called the [[tangent space]] of <math>M</math> at <math>p</math>. Vectors in <math>T_pM</math> are thought of as the vectors tangent to <math>M</math> at <math>p</math>. A [[Riemannian metric]] <math>g</math> on <math>M</math> assigns to each <math>p</math> a [[positive-definite]] inner product <math>g_p : T_pM \times  T_pM \to \mathbf R</math> in a smooth way. A smooth manifold <math>M</math> endowed with a Riemannian metric <math>g</math> is a [[Riemannian manifold]], denoted <math>(M,g)</math>.

Let <math>x^1,\ldots,x^n</math> denote the standard coordinates on <math>\mathbf R^n.</math> The Euclidean metric <math>g^\text{euc}</math> is given by
: <math>g^\text{euc} = (dx^1)^2 + \cdots + (dx^n)^2</math>.{{sfn|Lee|2018|p=12-13}}
Euclidean space is the Riemannian manifold <math>(\mathbf R^n,g^\text{euc})</math>.

In Euclidean space, all tangent spaces are canonically identified with each other via translation, so it is easy to move vectors from one tangent space to another. '''Parallel transport of tangent vectors''' is a way of moving vectors from one tangent space to another along a curve in the setting of a general Riemannian manifold. Note that while the vectors are in the tangent space of the manifold, they might not be in the tangent space of the curve they are being transported along.

An [[affine connection]] on a Riemannian manifold is a way of differentiating vector fields with respect to other vector fields. A Riemannian manifold has a natural choice of affine connection called the [[Levi-Civita connection]]. Given a fixed affine connection on a Riemannian manifold, there is a unique way to do parallel transport of tangent vectors.{{sfn|Lee|2018|p=105-110}} Different choices of affine connections will lead to different systems of parallel transport.

===Precise definition===

Let <math>M</math> be a manifold with an affine connection <math>\nabla</math>. Then a vector field <math>X</math> is said to be {{Anchor|ParallelVectorField}}'''parallel''' if for any vector field <math>Y</math>, <math>\nabla_YX=0</math>. Intuitively speaking, parallel vector fields have ''all their [[derivative]]s equal to zero'' and are therefore in some sense ''constant''. By evaluating a parallel vector field at two points <math>x</math> and <math>y</math>, an identification between a tangent vector at <math>x</math> and one at <math>y</math> is obtained. Such tangent vectors are said to be '''parallel transports''' of each other.

More precisely, if <math>\gamma:I\rightarrow M</math> is a [[curve|smooth curve]] parametrized by an interval <math>[a,b]</math> and <math>\xi\in T_xM</math>, where <math>x=\gamma(a)</math>, then a [[vector field]] <math>X</math> along <math>\gamma</math> (and in particular, the value of this vector field at <math>y=\gamma(b)</math>) is called the '''parallel transport of <math>\xi</math> along <math>\gamma</math>''' if
#<math>\nabla_{\gamma'(t)}X=0</math>, for all <math>t\in [a,b]</math>
#<math>X_{\gamma(a)}=\xi</math>.
Formally, the first condition means that <math>X</math> is parallel with respect to the [[pullback (differential geometry)|pullback connection]] on the [[pullback bundle]] <math>\gamma^* TM</math>. However, in a [[local trivialization]] it is a first-order system of [[linear differential equation|linear ordinary differential equations]], which has a unique solution for any initial condition given by the second condition (for instance, by the [[Picard–Lindelöf theorem]]).

The parallel transport of <math>X \in T_{\gamma(s)} M</math> to the tangent space <math>T_{\gamma(t)} M</math> along the curve <math>\gamma : [0,1] \to M</math> is denoted by <math>\Gamma(\gamma)_s^t X</math>. The map
: <math>\Gamma(\gamma)_s^t : T_{\gamma(s)} M \to T_{\gamma(t)} M</math>
is linear. In fact, it is an isomorphism. Let <math>\overline\gamma : [0,1] \to M</math> be the inverse curve <math>\overline\gamma(t) = \gamma(1-t)</math>. Then <math>\Gamma(\overline\gamma)_t^s</math> is the inverse of <math>\Gamma(\gamma)_s^t</math>.

To summarize, parallel transport provides a way of moving tangent vectors along a curve using the affine connection to keep them "pointing in the same direction" in an intuitive sense, and this provides a [[linear isomorphism]] between the tangent spaces at the two ends of the curve. The isomorphism obtained in this way will in general depend on the choice of the curve. If it does not, then parallel transport along every curve can be used to define parallel vector fields on {{mvar|M}}, which can only happen if the curvature of {{math|∇}} is zero.

A linear isomorphism is determined by its action on an [[Basis (linear algebra)#Ordered bases and coordinates|ordered basis]] or '''frame'''. Hence parallel transport can also be characterized as a way of transporting elements of the (tangent) [[frame bundle]] {{math|GL(''M'')}} along a curve. In other words, the affine connection provides a '''lift''' of any curve {{mvar|γ}} in {{mvar|M}} to a curve {{mvar|γ&#x303;}} in {{math|GL(''M'')}}.

===Examples===

The images below show parallel transport induced by the Levi-Civita connection associated to two different Riemannian metrics on the [[punctured plane]] <math>\mathbf R^2 \backslash \{0,0\}</math>. The curve the parallel transport is done along is the unit circle. In [[Polar coordinate system|polar coordinates]], the metric on the left is the standard Euclidean metric <math>dx^2 + dy^2 = dr^2 + r^2 d\theta^2</math>, while the metric on the right is <math>dr^2 + d\theta^2</math>. This second metric has a singularity at the origin, so it does not extend past the puncture, but the first metric extends to the entire plane.

{{multiple image
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| width     = 200
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| header = Parallel transports on the punctured plane under Levi-Civita connections
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|image1=Cartesian_transport.gif
|width1=200
|caption1=This transport is given by the metric <math>dr^2 + r^2 d\theta^2</math>.
|alt1=Cartesian transport

|image2=Circle_transport.gif
|width2=200
|caption2=This transport is given by the metric <math>dr^2 + d\theta^2</math>.
|alt2=Polar transport
}}

Warning: This is parallel transport on the punctured plane ''along'' the unit circle, not parallel transport ''on'' the unit circle. Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle. Since the first metric has zero curvature, the transport between two points along the circle could be accomplished along any other curve as well. However, the second metric has non-zero curvature, and the circle is a [[geodesic]], so that its field of tangent vectors is parallel.

===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).