Stiefel Manifold

The Stiefel manifold of Orthonormal $k$-frames in $\Bbb{R}^n$ is the collection of vectors ($v_1$, ..., $v_k$) where $v_i$ is in $\Bbb{R}^n$ for all $i$, and the $k$-tuple ($v_1$, ..., $v_k$) is Orthonormal. This is a submanifold of $\Bbb{R}^{nk}$, having Dimension $nk-(k+1)k/2$.

Sometimes the ``orthonormal'' condition is dropped in favor of the mildly weaker condition that the $k$-tuple ($v_1$, ..., $v_k$) is linearly independent. Usually, this does not affect the applications since Stiefel manifolds are usually considered only during Homotopy Theoretic considerations. With respect to Homotopy Theory, the two definitions are more or less equivalent since Gram-Schmidt Orthonormalization gives rise to a smooth deformation retraction of the second type of Stiefel manifold onto the first.

See also Grassmann Manifold