For example an infinite line and a circle. You can form a cross product which is an infinite tube.
What about a sphere and a plane? Can they be combined to form a manifold which is not a simple cross product of the two? What about higher dimensions such as $\mathbb{R}^4$ with $S_4$. Can they be combined so that the 8 dimensional manifold is not just the product of the two? Are there some examples.
I am thinking of the way, in group theory, SU(2) and SU(3) can be embedded in SU(5) which contains the cross product SU(2)xSU(3) but is not a simple cross product itself.
I wondered if open and closed spaces can be combined in non-trivial ways too?
(What got me thinking is that if the Universe is infinite and 4 dimensional and has some compact dimensions on a sphere, say, is the manifold always just the cross product of the open and closed spaces or can it be part of something more interesting?)