Briefly, is there an intuitive way to understand the relation $\mathbb{P}^n = \frac{U(n+1)}{U(1)\times U(n)}$ ? For instance, can one relate it to the usual definition of complex projective spaces as $\mathbb{P}^n = \left(\mathbb{C}^{n+1} \setminus \{0\}\right) / \mathbb{C}^* $ ?

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    $\begingroup$ Yes. To identify a space as $G/H$, you find a transitive action of $G$ on it with stabilizer $H$. The action of $U(n+1)$ on $\mathbb{P}^n$ is the obvious one coming from the interpretation of $\mathbb{P}^n$ as classifying lines in $\mathbb{C}^{n+1}$. $\endgroup$ – Qiaochu Yuan Nov 5 '18 at 23:18
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    $\begingroup$ If you write down the definition of $U(n+1)/(U(1) \times U(n))$ it boils down to look at the first column mod the action of the scalar $U(1)$ i.e it's exactly $\Bbb C^{n+1} \backslash \{0\}/ \Bbb C^*$ $\endgroup$ – Nicolas Hemelsoet Nov 5 '18 at 23:20

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