Editing Parallel transport (section)

===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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| 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.