Complex Stiefel manifold
= Complex Stiefel manifold
{title2=$V_k(\mathbb C^n)=U(n)/U(n-k)$}
The space of ordered orthonormal complex $k$-frames in $\mathbb C^n$, for $0\leq k\leq n$. It is a compact <smooth manifold> of real dimension $2nk-k^2$. Completing a frame to a unitary basis gives the displayed homogeneous-space description; $V_0$ is a point and $V_n(\mathbb C^n)$ is the <unitary group> $U(n)$.