Let's say we have a torus $T$ parametrized by $\varphi(\theta,\phi)=((a + b \cos \phi)\cos\theta, (a + b \cos \phi )\sin\theta, b \sin\phi)$ and $a>b>0$.

We construct a smooth tangent unit vector field $X$ on $T$ by differentiating $\varphi$ with respect to $\theta$ and normalizing it to obtain $X(\theta,\phi)=(-\sin(\theta),\cos(\theta),0)$.

Now $X$ looks pretty much like $(-y,x,0)$, but then normalized, so $\mathbf Y(x,y,z)=\left((x,y,z),\frac{1}{\sqrt{x^2+y^2}}(-y,x,0)\right)$ would be the non-parametrized version (which we will not use, but maybe it's relevant for the answer).

Question: Find a parallel tangent vector field $Z$ along the curve $\alpha : [0, 2\pi] \to T$ given by $\alpha(t) = \varphi(2t, 3t)$, non-zero and distinct from $X$.

Now I don't know how to tackle this type of problem, as I don't know how to construct parallel vector fields along any curve. The book I'm using only names a parallel vector field for the sphere (and just spontaneously names one, without explaining how it can be found).

So just to make a start, it wouldn't surpise me for some reason if I needed to use \begin{align*} \alpha'(t)=(-&3b\sin(3t)\cos(2t)-2(a+b\cos(3t))\sin(2t),\\ -&3b\sin(3t)\sin(2t)+2(a+b\cos(3t))\cos(2t),\\ &3b\cos(3t)) \end{align*}

Any help would be appreciated!


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