# Topological variations of the spherical shell

In 2D the space between two concentric circles (including border) is called an annulus. Topologically it is identical to the surface of a 3D cilinder.

Identifying points on the two concentric circles transforms the annulus into the surface of a 3D torus or a Klein surface (surface of a Klein bottle). This depends whether one identifies points lying on one half of a straigth line through the center of the circles, or on opposite halves of such straight lines (i.e. podal or anti-podal).

I am interested in the 3D generalisation of the annulus: the space lying between two concentric spheres (including the border spheres) or a spherical shell. What happens when one identifies points on the two spheres like in the 2D annulus case (podal or anti-podal)?

Is it related to the 3D surface of a 4D torus?
Is there a 3D generalization of a Klein surface?
How can I start to research this?

• It is an open shell, annulus if circles. Yes the identification between two pairs points is interesting. – William Elliot Jun 29 '18 at 20:45
• If you identified the entire radial segment with its antipodal segment, then you would have $[0,1]\times \Bbb P^2$, where $\Bbb P^2$ is the projective plane. But in your case just the endpoints are glued to their antipodes. – Paul Sinclair Jun 30 '18 at 13:17