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I am currently coding a small application that reproduces the transport of a vector along a geodesic on a 2D sphere.

Here's a capture of this application : capture

You can see as pink vectors the vectors of curvilinear coordinates and in cyan the transported vector.

The transport of vector along the geodesic (in yellow) is well reproduced from algorithmic and geometric point of view (with the using of Matrix rotation and angles computation).

Now I would like to check the validity of this transport from an analytical point of view with the equations of transport on geodesic.

For this, I am using the equations deduced from the definition of parallel transport :

$$\text{d}v^{\theta} = v^{\varphi}\sin\theta\,\cos\theta\,\text{d}\varphi\quad\quad (1)$$

and

$$\text{d}v^{\varphi} = -\cot\theta\,(v^{\theta}\text{d}\varphi + v^{\varphi}\text{d}\theta)\quad\quad (2)$$

In my simulation, I take as a convention that $e_{\theta}$ and $e_{\varphi}$ have a norm equal to 1, so I can write the vector transported like :

$$\vec{V}=V^{\theta}\,\vec{e_{\theta}}+V^{\varphi}\,\vec{e_{\varphi}}$$

and take : $$||\vec{e_{\theta}}|| = 1$$ and $$||\vec{e_{\varphi}}|| = 1$$

Actually, I wonder how to apply factors (on equations above (1) and (2)), factors that come from the definition of curvilinear basis vectors, i.e with Einstein notation :

$$\text{d}\vec{OM}=\dfrac{\partial\vec{OM}}{\partial y^{i}}\text{d}y^{i}$$

with the curvilinear basis vector $\vec{e_{i}}= \dfrac{\partial\vec{OM}}{\partial y^{i}}$

So, with this definition, I get for $(\vec{e_{\theta}},\vec{e_{\varphi}})$, starting from :

$$\vec{OM}=r\sin(\theta)\cos(\varphi)\vec{e_{x}}+r\sin(\theta)\sin(\varphi)\vec{e_{y}}+r\cos(\theta)\vec{e_{z}}$$

$$\vec{e_{\theta}}=r\cos(\theta)\cos(\varphi)\vec{e_{x}}+r\cos(\theta)\sin(\varphi)\vec{e_{y}}-r\sin(\theta)\vec{e_{z}}$$

$$\vec{e_{\varphi}}=-r\sin(\theta)\sin(\varphi)\vec{e_{x}}+r\sin(\theta)\cos(\varphi)\vec{e_{y}}$$

So the norms are equal to :

$$||\vec{e_{\theta}}|| = r$$ and $$||\vec{e_{\varphi}}|| = r\sin(\theta)$$

How to make the link between above norms of $(\vec{e_{\theta}},\vec{e_{\varphi}})$ and the coordinates that I am using in my code :

I mean, if it is necessary, how to apply these factors on equation (1) and (2) and mainly on which terms ($\text{d}v^{\theta}, \text{d}v^{\varphi}, v^{\theta}, v^{\varphi}$) ?

Thanks

UPDATE 1 :

I show you below the results that I get, the first figure represents $\theta$ and $\varphi$ during the transport and the second represents the total differential $Dv^{\theta}$ and $Dv^{\varphi}$ for small $\delta$ values (interval between 2 discrete values), i.e the following relations (3) and (4) :

$$\delta v^{\theta} - v^{\varphi}\sin\theta\,\cos\theta\,\delta\varphi=0\quad\quad (3)$$

with $$v^{\varphi}=v^{\varphi}_{i}$$ $$\delta v^{\theta} = v^{\theta}_{i+1}-v^{\theta}_{i}$$ $$\delta\varphi=\varphi_{i+1}-\varphi_{i}$$ $$\theta = \theta_{i}$$

and

$$\delta v^{\varphi} + \cot\theta\,(v^{\theta}\delta\varphi + v^{\varphi}\delta\theta)=0\quad\quad (4)$$

with $$v^{\varphi}=v^{\varphi}_{i}$$ $$v^{\theta}=v^{\theta}_{i}$$ $$\delta v^{\varphi} = v^{\varphi}_{i+1}-v^{\varphi}_{i}$$ $$\delta\varphi=\varphi_{i+1}-\varphi_{i}$$ $$\delta\theta=\theta_{i+1}-\theta_{i}$$ $$\theta = \theta_{i}$$

These 2 relations (3) and (4) must be equal to zero. I have taken a transported vector of norm equal roughly to 20.

Theta - Phi during transport Curves representing the total differential of $v^{Theta}$ and $v^{Phi}$

Do you think these results are consistent with theorical expectations ?

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    I'm not sure what you are trying to do. Parallel transporting a vector $\vec{V} = V^{\theta} \hat{e_\theta} + V^{\phi} \hat{e_\phi}$ along a geodesic doesn't do anything (the components are constant). The reason is that $\hat{e_{\theta}}$ is already parallel and $\hat{e_{\phi}}$ is orthogonal to $\hat{e_{\theta}}$ of unit length. Are you trying to calculate the parallel transport along an arbitrary curve? – 2017-01-28
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    In any case, this might be of use: http://www.physics.usu.edu/Wheeler/GenRel2013/Notes/Geodesics.pdf – 2017-01-28
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    :levap I just want to check if absolute differential of vector components $V^{i}$ is zero like the definition of parallel transport along a great circle (geodesic curves on sphere) say. My animation reproduces well the transport of vector cyan (see capture) and during this animation, the components $V_{\theta}$ and $V_{\varphi}$ are changing in tangent local basis (because local basis vectors are changing too during the transport, cf Affine connection). I try to check if these components are right compared to the equations (3) and (4), i.e which rule the value of $V_{\theta}$ and $V_{\varphi}$ – 2017-02-09
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    :levap as you can see on the last figure, I can't get exactly zero values for absolute differential of vector components $D\,V^{\theta}$ and $D\,V^{\varphi}$ with the definition : $D\,V^{i}=\nabla_{j}V^{i}\,dx^{j}$ with covariant derivative operator $\nabla$ – 2017-02-09

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