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?
This is trying to illustrate how the second space can be transformed to the first one, by turning the inner circle by 90 degees (and everything inbetween accordingly).