A metrically homogeneous space is a metric space $(X,d)$ such that for all points $p$ and $q$ in $X$, there exists an isometry $f$ such that $f(p) = q$. Does the sphere $S^n$ have this property?
odd dimension for considering as complex. Just take $f(x)=qp^{−1}x$ which would be the generalization of the rotation. If it even?