Parallel transport on a cardioid

[JonnyMaddox]

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) $X=\frac{dx^{i}}{dt}\frac{\partial}{\partial x^{i}}$ and a curve $x^{i}=x^{i}(t)$. 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

$x(t)=a(1+2\cos t + \cos 2t)$

$y(t)=a(2\sin t + \sin 2t)$

(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 $f$, but what function? A vector function? For example could I just take $f=r(\phi, \rho)= (\rho \cos \phi, \rho \sin \phi)$ (polar coordinates) and then $X=\frac{dx^{i}}{dt}\frac{\partial}{\partial x^{i}}= \frac{dx(t)}{dt}\frac{\partial r(\phi, \rho)}{\partial \phi}+\frac{dy(t)}{dt}\frac{\partial r(\phi,\rho)}{\partial \rho}$ ? I think this looks right since the $\frac{\partial}{\partial x^{i}}$ span the tangent space. Now how exactly does the condition for parallel transport in coordinates for this looks like?
The general formula is

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

(I know how to calculate the Levi-Civita connection with the metric,but I'm not sure about the rest)

 

[Evo]

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) $X=\frac{dx^{i}}{dt}\frac{\partial}{\partial x^{i}}$ and a curve $x^{i}=x^{i}(t)$. 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

$x(t)=a(1+2\cos t + \cos 2t)$

$y(t)=a(2\sin t + \sin 2t)$

(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 $f$, but what function? A vector function? For example could I just take $f=r(\phi, \rho)= (\rho \cos \phi, \rho \sin \phi)$ (polar coordinates) and then $X=\frac{dx^{i}}{dt}\frac{\partial}{\partial x^{i}}= \frac{dx(t)}{dt}\frac{\partial r(\phi, \rho)}{\partial \phi}+\frac{dy(t)}{dt}\frac{\partial r(\phi,\rho)}{\partial \rho}$ ? I think this looks right since the $\frac{\partial}{\partial x^{i}}$ span the tangent space. Now how exactly does the condition for parallel transport in coordinates for this looks like?
The general formula is

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

(I know how to calculate the Levi-Civita connection with the metric,but I'm not sure about the rest)

Sorry Johhny, is there anything else you can add or simplify if you still need an answer?

 

[lavinia]

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.