Manifold Structure of Spaces of Spherical Tight Frames

We consider the space FE k,n of all spherical tight frames of k vectors in the n–dimensional Hilbert space E (k > n), for E = R or E = C, and its orbit space GE k,n = FE k,n/O n under the obvious action of the group OE n of structure preserving transformations of E. We show that the quotient map FE k,n → GE k,n is a locally trivial fiber bundle (also in the more general case of ellipsoidal tight frames) and that there is a homeomorphism GE k,n → GE k,k−n. We show that GE k,n and FE k,n are real manifolds whenever k and n are relatively prime, and we describe them as a disjoint union of finitely many manifolds (of various dimensions) when when k and n have a common divisor. We also prove that FR k,n is connected (k ≥ 4) and FR n+2,n is connected, (n ≥ 2). The spaces GR 4,2 and GR 5,2 are investigated in detail. The former is found to be a graph and the latter is the orientable surface of genus 25.