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Thread: Parallel Transport

  1. #1 Parallel Transport 
    Forum Radioactive Isotope
    Join Date
    Feb 2009
    Vexpits gave us this link but I still have trouble with the method

    Suppose we choose 3 points on the surface of the Earth (or any sphere) that are 90 degrees at the apexes.

    Ie anywhere on the equator for the first point , a quarter revolution along the equator for the second and the North pole for the third.

    If we draw the triangle with these points the itinerary for parallel transporting is discontinuous at each apex.(and thus has me confused at these abrupt turns in the road)

    Can we generalize these sharp turns so that the apexes are replaced with smooth turnings? How will that work?

    Just as well?

    Also what happens if our itinerary starts at the NPole but returns to it in the form of a circle?Will that also allow one to measure the intrinsic curvature of the sphere?

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  3. #2  
    Moderator Moderator Markus Hanke's Avatar
    Join Date
    Nov 2011
    Have a look at my article on the topic of curvature, about half way down the page, it shows you what you are asking about:

    Manifolds and Curvature

    And no, it doesn’t make a difference whether the corners are sharp or round.

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