advanced_tools:parallel_transport

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advanced_tools:parallel_transport [2018/04/14 11:30] aresmarrero [Intuitive] |
advanced_tools:parallel_transport [2018/04/14 13:51] (current) theodorekorovin ↷ Page moved from advanced_notions:parallel_transport to advanced_tools:parallel_transport |
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<tabbox Intuitive> | <tabbox Intuitive> | ||

+ | To understand parallel transport imagine you're holding a long stick in your hand. You are then asked to walk a given path while keeping the stick straight all the time. What you are doing then is a parallel transport of the stick. | ||

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+ | The important thing is now that if you do this and walk a closed curve, i.e. you end up where you started, the final position of the stick will not be the same as the initial position. This happens no matter how good you are in keeping the stick straight. | ||

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+ | For example, consider two different paths around a ball as shown in the image below. | ||

+ | {{ :advanced_notions:curvature.png?nolink |}} | ||

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+ | If you follow the path on the left-hand side, where the position of the stick corresponds to the yellow arrows, the final position of the stick is not the initial position, since a ball is curved and not flat. | ||

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+ | ---- | ||

<blockquote>Parallel transport on a sphere can best be understood by imagining the sphere to be rolling on a flat | <blockquote>Parallel transport on a sphere can best be understood by imagining the sphere to be rolling on a flat | ||

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</note> | </note> | ||

- | <tabbox Why is it interesting?> | + | <tabbox Why is it interesting?> |

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- | + | The idea of parallel transport is important when we want to find out whether a given space is curved or not. | |

</tabbox> | </tabbox> | ||

advanced_tools/parallel_transport.1523698208.txt.gz · Last modified: 2018/04/14 09:30 (external edit)

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