I want to find the identifications of the quotient spaces for the following identifications of $U(n) \hookrightarrow U(n+1)$
Consider the inclusion given by $i_1$,
$i_1: A \rightarrow \begin{pmatrix} 1 & 0 & ... & 0\\ 0 & & & \\ \vdots & & A & \\ 0 & & & \end{pmatrix}$ which yields $U(n+1)/U(n) \cong \mathbb{S}^{2n-1}$
Consider the inclusion given by $i_2$,
$i_2: A \rightarrow \begin{pmatrix} (\det(A))^{-1} & 0 & ... & 0\\ 0 & & & \\ \vdots & & A & \\ 0 & & & \end{pmatrix}$ where we have $U(n+1)/U(n) \cong \mathbb{S}^1 \times \mathbb{CP}(n+1)$
I would like to know why in the first case it gives us $ \mathbb{S}^{2n - 1} $ and in the other case $ \mathbb{S}^{1} \times \mathbb{CP} (n-1) $ I understand that in the second case you have to $ i_2(U(n)) \subseteq SU(n + 1) $
$U(n+1)$ acts transitively on the 1-sphere in $\mathbf{C}^{n+1}$, with stabilizer $U(n)$ (embedded by $i_1$). This gives an identification of $U(n+1)/i_1(U(n))$ to rather the sphere $S^{2n+1}$.
Also it acts on the projective space $P_\mathbf{C}^n$, and also acts on the 1-sphere by $g\cdot z=\det(g)z$. For $n\neq 0$ this yields a transitive action on the product $P_\mathbf{C}^n\times S^1$, whose point stabilizer is $S(U(n)\times U(1))=i_2(U(n))$. This gives an identification of $U(n+1)/i_2(U(n)$ to $P_\mathbf{C}^n\times S^1$.