Let $m_l: S^{n-1} \stackrel{z \mapsto z^l}{\to} S^{n-1}$ be a map of spheres. Then it induces a map on $( S^{n-1})^k$, given by product. Note the Stiefel manifold $V_k(\mathbb{R^{n}})$, the collection of orthonormal $k$-frames in $\mathbb{R}^n$ is a closed subset of $( S^{n-1})^k.$
My question is: What is the induce map of $m_l$ on $V_k(\mathbb{R}^n)?$
Any help will be appreciated.