Uniqueness results for noncommutative spheres and projective spaces

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 $S^{N-1}\mathbb R\subset S^{N-1}{\mathbb R,}\subset S^{N-1}{\mathbb R,+}$,thenoncommutativeprojectivespaces$P^{N-1}\mathbb R\subset P^{N-1}\mathbb C\subset P^{N-1}+$, and the projective orthogonal quantum groups $P{O}_N\subset P{O}_N^{\ast }\subset P{O}_N^{+}$.