Complex Stiefel manifold ($V_k({\mathbb{C}}^n)=U(n)/U(n-k)$)

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 $U(n)$.

Use the complement bundle on a complex Stiefel manifold inductively. If the ring for $V_k$ is exterior with smallest positive degree $2n-2k+1$, the complement bundle has Euler degree $2(n-k)$, in which base cohomology vanishes. Its Euler class is consequently zero. The Gysin ring splitting for an odd-dimensional sphere bundle adds one exterior generator of degree $2(n-k)-1$. The initial base $V_0$ is a point. This gives the full integral cohomology ring and in particular its torsion-freeness. At $k=n$, the unitary group has Poincare polynomial ${\prod }_{j=0}^{n-1}(1+t^{2j+1})$.

The orthogonal complements of the universal frames define a complex vector bundle of rank $n-k$ inside the trivial ${\mathbb{C}}^n$ bundle. The orthogonal projector $I-{\sum }_i{v}_i{v}_i^{\ast }$ is smooth of constant rank, giving local bundle frames. Its underlying real bundle has canonical orientation and rank $2(n-k)$. Its sphere bundle is exactly $V_{k+1}({\mathbb{C}}^n)$, by appending the unit vector in the complement to the frame.