Manifold structure of spaces of spherical tight frames

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We consider the space Fᵉₖ,ₙ of all spherical tight frames of k vectors in real or complex n--dimensional Hilbert space Eⁿ, i.e. E=R or E=C, and its orbit space Gᵉₖ,ₙ=Fᵉₖ,ₙ/Oᵉₙ under the obvious action of the group Oᵉₙ of structure preserving transformations of Eⁿ. We show that the quotient map Fᵉₖ,ₙ -> Gᵉₖ,ₙ is a locally trivial fiber bundle (also in the more general case of ellipsoidal tight frames) and that there is a homeomorphism Gᵉₖ,ₙ -> Gᵉₖ,ₖ₋ₙ. We show that Gᵉₖ,ₙ and Fᵉₖ,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 Fʳₖ,₂ is connected (k >= 4) and Fʳₙ₊₂,ₙ is connected, (n >= 2). The spaces Gʳ₄,₂ and Gʳ₅,₂ are investigated in detail. The former is found to be a graph and the latter is the orientable surface of genus 25.