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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 scissor/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}$.

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 Chebychev net can form or essentially get defined by asymptotic hyperbolic geodesic zero normal curvature lines only, which is belied from present evidence.

Thanks in advance for any information/insight.

Geodesics on pseudosphere

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  • $\begingroup$ @ Lucas Lavoyer Thanks for the link. Can you please refer me to the equation in the cited link? $\endgroup$ – Narasimham Mar 6 '18 at 8:57

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