What is a parametric co-ordinate representation of a (non-orientable) Möbius strip of constant negative Gauss curvature in $ \mathbb R^ 3.$
As per MB_wiki
$$ (x,y,z) = ( R+s \cos(t/2)) \cos t \,,(R+ s \cos(t/2) )\sin t \,, s \sin(t/2) ) $$
The standard parametrization above gives variable but negative $K$ per Equation (12). Some parametric re-adjustment could impart to it a constant negative K, or so I believe.As far my thoughts on the problem go I could not succeed in setting up a valid parametrization keeping half angle trig relations intact. In two rotations the point should come back periodic. Other trial and errors did not succeed.. there ought to be a stronger approach. If I hit one in the meantime shall post it here.
Any one typical or at least one parametrization isometric to $K=-1/R^2$ is requested, as I am unable to find even one such example in references anywhere. I checked Struik, Guggenheimer, the French 3dcurves site &c.
If possible please indicate a link that discusses any such possibilities. If there is an indication why the question does not make full sense, that is also enough guidance at the moment..
Thanking in advance,
Regards