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If a cube is suspended in mid air with rubber wires inside a hollow glass sphere, its orientations realizes the sphere $S^3$, also called SU(2), which is the the double cover of SO(3). (Is this correct?)

If the glass sphere is swimming on water and rotating, the rotation axis of the glass sphere has two angles describing its orientation in space. Is this $S^5$ or is it $S^3 \times S^2$?

How can I see the difference?

KReiser
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Gina Martelli
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  • Looks like the configuration space is a fibre bundle with base $S^2$ and fibre $S^3$. Can $S^5$ be such a fibre bundle? I suspect not. Must such a fibre bundle be $S^2\times S^3$? I'm not sure. – Angina Seng Apr 05 '20 at 19:48
  • Any principal bundle over $S^2$ with fibre $SU(2)$ is trivial, as $SU(2)$ is simply-connected. If you instead take the fibre bundle to have structure group $SO(3)$, then there are two non-isomorphic bundles, one trivial, one not. It's not clear to me that the second one is a 5-sphere, but I can't rule it out with doing a calculation. – theHigherGeometer Apr 05 '20 at 22:33

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