IJPAM: Volume 28, No. 2 (2006)

MANIFOLD STRUCTURE OF SPACES OF
SPHERICAL TIGHT FRAMES

Ken Dykema

, Nate Strawn

$^{1,2}$ Department of Mathematics
Texas A&M University
College Station, TX 77843-3368, USA

e-mail: Ken.Dykema@math.tamu.edu

e-mail: natestrawn@neo.tamu.edu

Abstract.We consider the space $\Fc^\Eb_{k,n}$ of all spherical tight frames of vectors in the -dimensional Hilbert space $\Eb^n$ (), for $\Eb=\Reals$ or $\Eb=\Cpx$ , and its orbit space $\Gc^\Eb_{k,n}=\Fc^\Eb_{k,n}/\Oc^\Eb_n$ under the obvious action of the group $\Oc^\Eb_n$ of structure preserving transformations of $\Eb^n$ . We show that the quotient map $\Fc^\Eb_{k,n}\to\Gc^\Eb_{k,n}$ is a locally trivial fiber bundle (also in the more general case of ellipsoidal tight frames) and that there is a homeomorphism $\Gc^\Eb_{k,n}\to\Gc^\Eb_{k,k-n}$ . We show that $\Gc^\Eb_{k,n}$ and $\Fc^\Eb_{k,n}$ are real manifolds whenever and are relatively prime, and we describe them as a disjoint union of finitely many manifolds (of various dimensions) when when and have a common divisor. We also prove that $\Fc^\Reals_{k,2}$ is connected ( $k\ge4$ ) and $\Fc^\Reals_{n+2,n}$ is connected ( $n\ge2$ ). The spaces $\Gc^\Reals_{4,2}$ and $\Gc^\Reals_{5,2}$ are investigated in detail. The former is found to be a graph and the latter is the orientable surface of genus .