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?

  • $\begingroup$ It is an open shell, annulus if circles. Yes the identification between two pairs points is interesting. $\endgroup$ – William Elliot Jun 29 '18 at 20:45
  • $\begingroup$ 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. $\endgroup$ – Paul Sinclair Jun 30 '18 at 13:17


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).

  • $\begingroup$ Is this also true in 3D? $\endgroup$ – Gerard Jul 2 '18 at 9:18

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