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Parallel transport on a cardioid

  1. Sep 23, 2014 #1
    Hi guys,
    I want to calculate an explicit example of a vector parallel transported along a cardioid to see what happens. Maybe someone could help me with that since no author of any book or pdf on the topic is capable of showing a single numerical example.

    So we need a vector field on a manifold (which is the cardioid itself) [itex]X=\frac{dx^{i}}{dt}\frac{\partial}{\partial x^{i}}[/itex] and a curve [itex]x^{i}=x^{i}(t)[/itex]. My problem is, I'm not sure how to make up a curve + vector field on a manifold. Let's take the parametrization of the cardioid in Cartesian coordinates as

    [itex]x(t)=a(1+2\cos t + \cos 2t)[/itex]

    [itex]y(t)=a(2\sin t + \sin 2t)[/itex]

    (I think this could be written in polar coordinates which would make more sense, but I'm not sure what happens there)

    So I think this should be the curve on which the vector is transported. Now I'm not sure how to make up the vector field. For the vector field I also need a function [itex]f[/itex], but what function? A vector function? For example could I just take [itex]f=r(\phi, \rho)= (\rho \cos \phi, \rho \sin \phi)[/itex] (polar coordinates) and then [itex]X=\frac{dx^{i}}{dt}\frac{\partial}{\partial x^{i}}= \frac{dx(t)}{dt}\frac{\partial r(\phi, \rho)}{\partial \phi}+\frac{dy(t)}{dt}\frac{\partial r(\phi,\rho)}{\partial \rho}[/itex] ? I think this looks right since the [itex]\frac{\partial}{\partial x^{i}}[/itex] span the tangent space. Now how exactly does the condition for parallel transport in coordinates for this looks like?
    The general formula is

    [itex]\frac{\partial X^{\mu}}{dt}+ \Gamma^{\mu}_{v\lambda} \frac{\partial x^{v}(c(t))}{dt}X^{\lambda}=0[/itex]

    (I know how to calculate the Levi-Civita connection with the metric,but I'm not sure about the rest)
     
  2. jcsd
  3. Nov 15, 2014 #2

    Evo

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    Staff: Mentor

    Sorry Johhny, is there anything else you can add or simplify if you still need an answer?
     
  4. Nov 17, 2014 #3

    lavinia

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    Science Advisor

    The cardiod is a curve in the plane and I guess you want to parallel translate a vector along it using the standard inner product on the plane.

    - what do you get for the covariant derivative in the plane?

    Once you have that all else follows quickly.
     
    Last edited: Nov 17, 2014
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