<s>[INST] Loxodromics (curves with fixed angle with meridians) of a revolution surface I am trying to find the curves with a fixed angle $\phi$ with meridians for the revolution surface given by the revolution of the graph of $z=\frac{1}{x}$ around the $z$ axis. The surface is easily parametrized as
$$(x,\theta)\mapsto(x \cos \theta, x\sin \theta,\frac{1}{x}) \qquad x\in \mathbb R_{>0},\theta \in (0,2\pi)$$
Now, some tedious computations (I hope they are correct) show that the first and the second fundamental form for this parametrization are
$$G_{I}=\begin{pmatrix}1+\frac{1}{x^4} &0 \\ 0 & x^2 \end{pmatrix} \qquad G_{II}=\frac{1}{\sqrt{x^4+1}}\begin{pmatrix}\frac{1}{x} & 0 \\ 0 & -x \end{pmatrix}$$
Recall that the angle $\phi$ between two $C^1$ curves $\gamma_1,\gamma_2$ is given by
$$\cos \phi=\frac{\gamma_1'^TG_I \gamma_2'}{\sqrt{\gamma_1'^T G_I \gamma_1'}\sqrt{\gamma_2'^TG_I\gamma_2'}}$$
Since we are looking for curves $(x(t),\theta(t))^T$ with constant angle with meridians, we want that
$$\frac{\begin{pmatrix}1 & 0 \end{pmatrix}\begin{pmatrix} 1+\frac{1}{x^4(t)} &0 \\ 0 & x^2(t)\end{pmatrix}\begin{pmatrix} x'(t) \\ \theta'(t) \end{pmatrix}}{\sqrt{\begin{pmatrix} x(t) & \theta(t)\end{pmatrix}\begin{pmatrix}1+\frac{1}{x^4(t)} &0 \\ 0 & x^2(t) \end{pmatrix}\begin{pmatrix} x'(t)\\ \theta'(t) \end{pmatrix}}\sqrt{\begin{pmatrix} 1 & 0\end{pmatrix}\begin{pmatrix}1+\frac{1}{x^4(t)} &0 \\ 0 & x^2(t) \end{pmatrix}\begin{pmatrix} 1 \\ 0\end{pmatrix}}}=K$$
which, dropping the dependency on $t$, yields the differential equation
$$x'\left (1+\frac{1}{x^4}\right )=K\sqrt{1+\frac{1}{x^4}}\sqrt{x'^2\left ( 1+\frac{1}{x^4}\right)+x^2\theta'^2}$$
Not looking very good. Through a few manipulations, I brought this to the form
$$x'(t)=K\sqrt{x'^2+\frac{x^6\theta'^2}{x^4+1}}$$
Since this is an exercise from an exam, I would expect the solution to have a nice closed form, but maybe I am mistaken. 
Is there any elementary way of solving such differential equation or of finding an explicit equation for the loxodromics?
 [/INST] Well, actually you can also use the vector $(0,1)$, since the tangent vectors to the coordinate lines $x$ and $\theta$ are orthogonal at each point. Then your curve measures a constant angle $\alpha_0$ with the tangent $x-$vector if and only if it also measures a constant angle $\pi/2 - \alpha_0$ with the tangent $\theta-$vector, so you can go this way too. But anyway, the result is going to be virtually the same. You end up with the dot product expression (i.e. the dot product formula $(a \cdot b) = |a||b|\cos{\alpha_0} \,\,$):
\begin{align} (1,0) \,\,  G_I(x) \, \, (x',\theta')^T &= \sqrt{\Big((1,0) \,\, G_I(x) \,\, (1,0)^T\Big)} \sqrt{\Big((x',\theta') \,\, G_I(x) \,\, (x',\theta')^T\Big)} \, \cos{(\alpha_0)} \\
\left(1+ \frac{1}{x^4}\right) x' &= \cos{(\alpha_0)}\sqrt{\left(1+ \frac{1}{x^4}\right)}  \sqrt{\left(1+ \frac{1}{x^4}\right)(x')^2 + x^2 (\theta')^2}  
\end{align}
Notice that all expressions, apart from the derivative $\theta'$, depend only on the variable $x$! This is not a coincidence as the surface is a surface of revolution and is therefore rotatiationally invariant with respect to rotations around the $z$ axis, which in the $x,\theta$ coordinates manifest themselves as translations of the $\theta$ coordinate, i.e. $\theta \mapsto \theta + c$. Consequently, when you have one curve on the surface of constant angle $\alpha_0$ with respect to the $x-$coordinate curves, if you rotate it around $z$, you get again a curve on the surface of the same constant angle $\alpha_0$ with respect to the $x-$coordinate curves. Therefore, your equations for the curves of angle $\alpha_0$ should be invariant with respect to the symmetries $\theta \mapsto \theta + c$, which means that the equations do not depend explicitly on $\theta$. Thus, we can simply write the curve as $\theta = \theta(x)$, i.e. it is parametrized as $x \mapsto (x, \theta(x))$, so $x' = 1$ and $\theta' = \frac{d\theta}{dx}$. So the equation becomes
\begin{align} 
\left(1+ \frac{1}{x^4}\right) &= \cos{(\alpha_0)}\sqrt{\left(1+ \frac{1}{x^4}\right)}  \sqrt{\left(1+ \frac{1}{x^4}\right) + x^2 \Big(\frac{d\theta}{dx}\Big)^2};\\
 \sqrt{1+ \frac{1}{x^4}} &= \cos{(\alpha_0)} \sqrt{\left(1+ \frac{1}{x^4}\right) + x^2 \Big(\frac{d\theta}{dx}\Big)^2};\\
 {1+ \frac{1}{x^4}} &= \cos^2{(\alpha_0)}\left( 1+ \frac{1}{x^4}+ x^2 \Big(\frac{d\theta}{dx}\Big)^2\right);\\
\frac{1}{\cos^2{(\alpha_0)}}\left({1+ \frac{1}{x^4}}\right) &= 1+ \frac{1}{x^4}+ x^2 \Big(\frac{d\theta}{dx}\Big)^2;\\
 x^2 \Big(\frac{d\theta}{dx}\Big)^2 &= \left(\frac{1}{\cos^2{(\alpha_0)}} - 1\right)\left(1+ \frac{1}{x^4}\right) \\
 \Big(\frac{d\theta}{dx}\Big)^2 &= \tan^2{(\alpha_0)}\,\frac{1}{x^2}\left(1+ \frac{1}{x^4}\right) = \tan^2{(\alpha_0)}\,\left(\frac{x^4+1}{x^6}\right);\\
\frac{d\theta}{dx} &= \pm \tan{(\alpha_0)}\,\left(\frac{\sqrt{x^4+1}}{x^3}\right);
\end{align}
So as you can see, there is almost no actual differential equation, the problem is solved by simply explicitly integrating a derivative
\begin{align}
\theta(x) &= \theta_0 \pm k_0 \,\int \frac{\sqrt{x^4+1}}{x^3}dx,
\end{align} where $k_0 = \tan{\alpha_0}$. 
</s>