= Complement bundle on a complex Stiefel manifold
{title2=$E_{(v_1,\ldots,v_k)}=\langle v_1,\ldots,v_k\rangle^\perp$}
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 $I-\sum_i v_iv_i^*$ 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.
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