$U(n)/U(n-1)$ as homogeneous space

Question

How can I prove that the quotient $$\frac{U(n)}{U(n-1)} \simeq S^{2n-1}$$ (where $U(n)$ is the unitary group). Is it correlated with the theory of homogeneous spaces?

Answer 1

I think you meant $S^{2n-1}$. Yes, this is the beginning of the theory of homogeneous spaces: if a Lie group $G$ acts transitively by homeomorphisms on a topological set $X$, then $X$ is homeomorphic to the quotient of $G$ by the stabilizer of some marked point $o\in X$. If $X$ is a smooth manifold, you can replace "homeo" with "diffeo" on both occasions.

Here $S^{2n-1}=\{(z_1,\dots,z_n)\in \mathbb C^n: \sum |z_j|^2=1\}$. The unitary group $U(n)$ acts on $\mathbb C^n$ by isometries, and therefore preserves $S^{2n-1}$. It is easy to see that the action is transitive: any unit vector is a part of some orthonormal basis, and any two orthonormal bases are related by an element of $U(n)$. Finally, the stabilizer of the point $o=(1,0,\dots,0)$ consists of the matrices $A\in U(n)$ of the form $$ A = \begin{pmatrix} 1 & 0 & \dots & 0 \\ 0 & \\ \vdots & & \ddots\\ 0& \end{pmatrix} $$ The space markes with $\ddots$ is a unitary matrix of size $(n-1)$. This identifies the stabilizer of $o$ with $U(n-1)$.