  It is known that, under strong combinatorial axioms, $O_N\subset O_N^*\subset
O_N^+$ are the only orthogonal quantum groups. We prove here similar results
for the noncommutative spheres $S^{N-1}_\mathbb R\subset S^{N-1}_{\mathbb
R,*}\subset S^{N-1}_{\mathbb R,+}$, the noncommutative projective spaces
$P^{N-1}_\mathbb R\subset P^{N-1}_\mathbb C\subset P^{N-1}_+$, and the
projective orthogonal quantum groups $PO_N\subset PO_N^*\subset PO_N^+$.
