Volume of geodesic balls in the complex Stiefel manifold

Abstract

Volume estimates of geodesic balls in Riemannian manifolds find many applications in coding and information theory. This paper computes the precise power series expansion of volume of small geodesic balls in a complex Stiefel manifold of arbitrary dimension. The volume result is employed to bound the minimum distance of codes over the manifold. An asymptotically tight characterization of the rate-distortion tradeoff for sources uniformly distributed over the surface is also provided.