I'm here trying to recognize an (n-1)-dimensional manifold in $R^n$ whose structure can be sumarized as Interval times union of two codimension 2 spheres that meet a common cod-2 sphere at one end of the interval, and another common cod-2 sphere at the other end. My guess is it is a torus.
My motivation is, let's think of a standard 2-torus in $R^3$ horizontally placed. By fixing descendant values for z, we get (from top to bottom) a circle (codimension two sphere), union of two circles (likewise) until it hits the lower critical value again in one sphere.
Having this in mind, my guess is my manifold is a torus since it has that exact same structure but I'm not 100% certain...
All help is appreciated, thanks in advance.
The manifold is $S^{n-2}_r \times S^1$ also known as Dupin cyclide