= Real rotation orbits on a complex unit quadric
{title2=$M_r\cong SO(3)\ (r>0),\quad M_0\cong S^2$}
For $z=x+iy\in\mathbb C^3$ satisfying $z^Tz=1$, the real and imaginary parts obey $|x|^2-|y|^2=1$ and $x\cdot y=0$. Simultaneous real rotations preserve $r=|y|$. At $r=0$ the <group orbit> is the unit sphere, with <stabilizer subgroup> $SO(2)$. At each $r>0$, the normalized pair $(x,y)$ is an oriented two-frame, giving a free transitive <SO(3) group> action. The <group orbit> space is $[0,\infty)$, and the whole quadric is diffeomorphic to the <tangent bundle> of $S^2$. Its varying Hermitian norm prevents it from being a single <SU(3)> <group orbit>.
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