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Geodesic transport of a small 2D surface on a 3D manifold.

  1. Aug 16, 2015 #1
    I've just read and I think I have understood the following result :
    If we were to geodesically transport all points of a small 2D surface, so small that it would be flat for all purposes, in a direction vertically above it, and if this surface belongs in an arbitrary 3D manifold, then in general both its area and its shape will change. It is impossible to achieve a change in shape without its area to dwindle or expand simultaneously. And this is a property of the so-called Weyl tensor.
    I can somehow visualize how this is happening without recourse to advanced maths, but what a great result this is!
    In a 4D manifold, it is also true that the object can retain its volume even if the tidal forces (curvature) change its shape, but not in any lower dimension than 4.
    I have nothing more to ask, just to verify if this is true. What a great result! If it is true, it is a moment of revelation and enlightening for me.
  2. jcsd
  3. Aug 16, 2015 #2


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    Gold Member

    Yes, but where did you read it ?
  4. Aug 16, 2015 #3
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