Stiefel Manifold - Topology

Topology

Let F stand for R, C, or H. The Stiefel manifold V_k(Fn) can be thought of as a set of n × k matrices by writing a k-frame as a matrix of k column vectors in Fn. The orthonormality condition is expressed by A*A = 1 where A* denotes the conjugate transpose of A and 1 denotes the k × k identity matrix. We then have

The topology on V_k(Fn) is the subspace topology inherited from Fn×k. With this topology V_k(Fn) is a compact manifold whose dimension is given by