Editing Riemannian manifold (section)

=== Parallel transport ===
{{Main|Parallel transport}}
[[File:Parallel transport sphere2.svg|thumb|right|Parallel transport of a tangent vector along a curve in the sphere.]]

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]] is a way of moving vectors from one tangent space to another along a curve in the setting of a general Riemannian manifold. Given a fixed connection, there is a unique way to do parallel transport.{{sfn|Lee|2018|pp=105–110}}

Specifically, call a smooth vector field <math>V</math> along a smooth curve <math>\gamma</math> ''parallel along <math>\gamma</math>'' if <math>D_t V = 0</math> identically.{{sfn|Lee|2018|p=105}} Fix a curve <math>\gamma : [0,1] \to M</math> with <math>\gamma(0) = p</math> and <math>\gamma(1) = q</math>. to parallel transport a vector <math>v \in T_pM</math> to a vector in <math>T_qM</math> along <math>\gamma</math>, first extend <math>v</math> to a vector field parallel along <math>\gamma</math>, and then take the value of this vector field at <math>q</math>.

The images below show parallel transport induced by the Levi-Civita connection associated to two different Riemannian metrics on the [[punctured plane]] <math>\mathbb R^2 \smallsetminus \{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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| 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.