 Complex Stiefel manifold

The space of ordered orthonormal complex $k$-frames in ${\mathbb{C}}^n$, for $0\le k\le n$. 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 $U(n)$.