  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.
