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?