Homogeneous space and nice manifolds

Question

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.

Do we have similar nice results for the following: $$ U(n+1)/U(n) \simeq? $$ and $$ SU(n+1)/SU(n) \simeq? $$ $$ Sp(n+1)/Sp(n) \simeq? $$

Answer 1

There are all spheres. The relevant technical lemma is the follow:

Suppose $G$ is a compact Lie group which acts smoothly and transitively on a manifold $M$. Let $p\in M$ and set $G_p = \{g\in G: gp = p\}$. Then $G/G_p$ is canonically diffeomorphic to $M$ via the diffeomorphism $g G_p\mapsto gp$.

For $G = U(n+1)$ or $G = SU(n+1)$, the relevant $M$ is the the sphere $S^{2n+1}\subseteq \mathbb{C}^{n+1}$, where the $G$ action is given by usual matrix multiplication on vectors in $\mathbb{C}^{n+1}$. For $G = Sp(n+1)$, $M = S^{4n+3}\subseteq \mathbb{H}^{n+1}$.

One must verify that the $G$ action is transitive, and then compute $G_p$ for a single $p$.

Technically, the notation $U(n+1)/U(n)$, etc, is ambiguous until one specifies an embedding $U(n)\rightarrow U(n+1)$. Fr the examples at hand, it is often the case (but not always the case) that there is a unique embedding $G_p\rightarrow G$, up to conjugacy. Conjugate embeddings always give rise to diffeomorphic manifolds: $G/G_p\cong G/gG_p g^{-1}$.

One the other hand, the two embeddings of $U(1)$ into $U(2)$ given by $z\mapsto \operatorname{diag}(z,1)$ or $\operatorname{diag}(z,z)$ give rise to homogeneous spaces which are not even homotopy equivalent. One has an analogous result for $Sp(1)\rightarrow Sp(2)$. But these are the only exceptions: e.g. $U(2)$ embeds into $U(3)$ in only way one, up to conjugacy.