Is it true that Sine-Gordon is satisfied for geodesics on the central Pseudosphere ( rotated surface of Tractrix)? If so, please cite text-book or article references. Searching for it in references suggested by Lucas Lavoyer
Chebychev scissorde/diamond cell Net for hyperbolic lines on a Beltrami pseudosphere are given by:
$$ \alpha''(s)=sin ( \alpha) $$ where $\alpha$ = $2 \psi$,$\psi$ is angle between geodesic and v= constant parameter lines.
I just discovered the above to be true but to check old literature an access to thorough academic base is lacking.
( Sine-Gordon ChebychevNet for asymptotic lines are well known, but not referring to it now).
I am adding Mathematica generated image and relation for polar plot in this finding, $( r,\theta, z) $ are cylindrical co-ordinates:
$$ (\theta/r_{oH})^2 = 1/r_{oE}^2 - 1/r^2 $$
where $ r_{oE} $ is the minimum (geodesic lines are orthogonal to meridian here, it is the Clairaut's constant $ r*sin\psi $ radius and $r_{oH}$ is maximum pseudospherical cuspidal radius at equator. These are the invariants in elliptic and hyperbolic branches of non-euclidean geometry that we are able to see here together on this pseudospherical surface of revolution. Image verifies circumferential disposition at minimum radius before going to cuspidal equator radius $r_{oH}$
If $\psi_{min}$ is angle where the geodesic meets cuspidal equator, $$ sin \psi_{min}= r_{oE} /r_{oH}$$
Derivation is simple and straightforward. Dropping (s) as for pure functions of arc length s.
Liouville's formula for a geodesic on surface of revolution:
$ \psi'$ =- $ sin\psi$ $sin\phi /r $ ; (1)
For a tractrix meridian : $ 1/r_{oH} = sin \phi /r $ ; (2)
plug above into (1) to get $ \psi' =-sin \psi/ r_{oH} $ ; (3)
Differentiate the above and plug into the same from above for $ \psi'$ to get
$ \psi''$= $sin \psi$ $cos \psi$ $ /r_{oH}^2$ ; $ 2 \psi = \alpha $ ; (4)
yields Sine-Gordon $ \alpha''= sin ( \alpha)/r_{oH}^2 $ for cusped pseudosphere radius $ r_{oH}$.
EDIT1:
So why is it that the nature of geodesics whether hyperbolic negative or elliptic Riemannian positive, never influences Sine-Gordon?
Earlier I was under the impression that a Chebychev net can form or essentially get defined by asymptotic hyperbolic geodesic zero normal curvature lines only, which is found untrue from present evidence.
Thanks in advance for any information/insights including any non-rotational surfaces in $ \mathbb R^3.$
