Manifold structure of spaces of spherical tight frames

TL;DR

This paper studies the geometric and topological structure of spaces of spherical tight frames, revealing their manifold properties, fiber bundle structure, and connectivity, with detailed analysis in specific low-dimensional cases.

Contribution

It establishes the fiber bundle structure of the space of spherical tight frames and characterizes their topological types, including homeomorphisms and manifold decompositions, especially when k and n are coprime.

Findings

F^E_{k,n} -> G^E_{k,n} is a locally trivial fiber bundle.

G^E_{k,n} is homeomorphic to G^E_{k,k-n}.

F^R_{k,2} is connected for k >= 4.

Abstract

We consider the space F^E_{k,n} of all spherical tight frames of k vectors in real or complex n--dimensional Hilbert space E^n, i.e. E=R or E=C, and its orbit space G^E_{k,n}=F^E_{k,n}/O^E_n under the obvious action of the group O^E_n of structure preserving transformations of E^n. We show that the quotient map F^E_{k,n} -> G^E_{k,n} is a locally trivial fiber bundle (also in the more general case of ellipsoidal tight frames) and that there is a homeomorphism G^E_{k,n} -> G^E_{k,k-n}. We show that G^E_{k,n} and F^E_{k,n} are real manifolds whenever k and n are relatively prime, and we describe them as disjoint unions of finitely many manifolds (of various dimensions) when when k and n have a common divisor. We also prove that F^R_{k,2} is connected (k >= 4) and F^R_{n+2,n} is connected, (n >= 2). The spaces G^R_{4,2} and G^R_{5,2} are investigated in detail. The former is found to be a graph and the latter is the orientable surface of genus 25.