Stiefel manifold

Remarks

Special cases: $V_1(\R^n) = S^{n-1}$, and $V_n(\R^n)$ is the orthogonal group $O(n)$. For $V_2(\R^3)$ the dimension is $6 - 3 = 3$, and indeed $(v_1, v_2) \mapsto (v_1, v_2, v_1 \times v_2)$ identifies it with the rotation group $SO(3)$.

Stiefel manifolds give standard fiber bundles. The map $V_k(\R^n) \to S^{n-1}$, $(v_1, \ldots, v_k) \mapsto v_1$, is a fiber bundle with fiber $V_{k-1}(\R^{n-1})$. The map $V_k(\R^n) \to G_k(\R^n)$ that sends a frame to the subspace it spans is a principal $O(k)$-bundle over the Grassmannian. Using these bundles one shows that $V_k(\R^n)$ is $(n - k - 1)$-connected; as $n \to \infty$, the bundle $V_k(\R^\infty) \to G_k(\R^\infty)$ becomes the universal bundle for $O(k)$.