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Equation for parallel transport involving sectional curvature

  1. Jan 10, 2014 #1
    This is an expression I came across in a paper I am going through. It involves an expression for the parallel transport of a tangent vector taking into consideration the sectional curvature of simply connected space-forms in [itex] \mathbb R^4 [/itex]. I have not been able to derive it.The equation simply states that
    [itex]P^{t}(v) = -\epsilon g(t)p+f(t)v[/itex]

    where [itex] \epsilon = 1, 0 ,-1 [/itex] depending on whether the space-form in question is the sphere, Euclidean plane and Hyperbolic plane [itex] \mathbb S^3 , \mathbb E^3, \mathbb H^3 [/itex] respectively and the functions are

    [itex] f(t) = 1, g(t)=t [/itex] for [itex] \mathbb E^3 [/itex]
    [itex] f(t) = \cos t, g(t)= \sin t [/itex] for [itex] \mathbb S^3 [/itex]
    [itex] f(t) = \cosh t, g(t)= \sinh t [/itex] for [itex] \mathbb H^3 [/itex]

    Can someone please tell me how the above can be explicitly computed without taking recourse to Lie Groups. A reference where this is worked out in detail would also be welcome. I have included the paper link. Thanks http://www.mathnet.or.kr/mathnet/thesis_file/BKMS-50-4-1099-1108.pdf
     
  2. jcsd
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