Is the following parameterized surface homeomorphic to a Möbius Band?
a=1;
g1=ParametricPlot3D[{u Cos[t],(a-u) Cot[t],u Sin[t]},{t,.001,Pi-.001},{u,.8,1.2},Mesh->{30,10},PlotStyle->{Yellow}]
g2=ParametricPlot3D[{u Cos[t],(a-u) Cot[t],u Sin[t]},{t,-.001,-Pi+.001},{u,.8,1.2},Mesh->{30,10},PlotStyle->{Yellow}]
Show[{g1,g2},PlotRange->{-1.15,1.15},Boxed->False,Axes->None]
How can the $y-$ coordinate be modified to obtain Gauss curvature $-1/a^2 $ everywhere constant?
As it is $K$ has this value only at start points $t= (0,\pi).$