In constructing a manifold for neutron decay cosmology the following surface evolved and unless I am mistaken this is a single sided closed surface but is NOT a Klein bottle.
Shirley’s Surface
$$\left(\cos\left(\frac{u}2\right)\cos\left(\frac{v}2\right),\,\cos\left(\frac{u}2\right)\sin\left(\frac{v}2\right),\,\frac{\sin(u)}2\right), 0 \le u \le 2\pi,\,0 \le v \le 2\pi$$
Notice that $2\pi$, a full rotation (Tau), only gets one half way over the surface. Am I incorrect in that is is a one sided closed surface? If so, the only one sided surface I can find mention of is Klein bottle, so next question is “Is this just a contorted Klein bottle?” But I don’t know if there is mathematical way to prove that or not.
This is why I reach out to community.
In my T.O.E this is why electron half spin. One expressed orbit on this side of temporal membrane and an internalized orbit as positron on other side of temporal membrane.
If the character of this shape has previously gone unrecognized, given that it will become an extremely important form in future astrophysics and with the application of Neutron Decay Cosmology, I claim privilege to name this surface “Shirley’s Surface”.
I plotted your equations with mathematica and this is what it looks like:
If you increase the domain to get the whole thing, then yes, it is an image of a Klein bottle.