We know that $$ O(n+1)/O(n) \simeq SO(n+1)/SO(n) \simeq S^n, $$ based on the result of homogeneous space.
Also $$ PO(n+1)/PO(n) \simeq P^n, $$ $P^n$ is the projective space.
These are in some sense spheres.
If we embed the real projective space $\mathbb{C}P^n$ into $\mathbb{R}P^{n+1}$, we may be able to define the quotient space
$$ {\mathbb{R}P^{n+1}}/{\mathbb{R}P^{n}} \simeq ? $$
Do we have simpler expressions for the above "manifolds" or "quotient space"?
Homotopy group $\pi_i({\mathbb{R}P^{n+1}}/{\mathbb{R}P^{n}})=?$
My Attempt: Note that ${\mathbb{R}P^{n}}$ is diffeomorphic to the submanifold of $R^{(n+1)^2}$ consisting of all symmetric $(n+1) \times (n+1)$ matrices of trace 1 that are also idempotent linear transformations. , so for ${\mathbb{R}P^{2}}\simeq SO(3)$, ${\mathbb{R}P^{1}}\simeq S^1 \simeq SO(2)$ and ${\mathbb{R}P^{0}}\simeq 0$, so $$\pi_i({\mathbb{R}P^{2}}/{\mathbb{R}P^{1}})=\pi_i(SO(3)/SO(2))=?,$$ $$\pi_i({\mathbb{R}P^{1}}/{\mathbb{R}P^{0}})=\pi_i(S^1),$$ Also the $\pi_i({\mathbb{R}P^{n}})$ can be found here.