Any odd dimensional sphere $S^{2n+1}$ can be expressed as an homogenous space of $SU(n+1)$ by $S^{2n+1} \simeq SU(n+1)/SU(n)$. Any even dimensional sphere $S^{2n}$ sphere can be expressed as an homogeneous space of $SO(n)$ according to $S^{2n} \simeq SO(n)/SO(n-1)$. Can we make for a switch here? In explicit, can one realize the odd spheres as $SO(n)$ homogeneous spaces, and can one realize the even spheres as $SU(n)$ homogeneous spaces.
2026-03-25 23:36:27.1774481787
Spheres as Homogeneous Spaces
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The first part of the question has a positive answer, since we have $$ S^n\cong SO(n+1)/SO(n) $$ also for odd $n$. For the case $SU(n)$ this is not possible in this way, however it might be realized in a different way, e.g., $$ S^6\cong G_2/SU(3), $$ where $n=6$ is even.