The configuration space of SU(N) gauge theory is restricted to orbits with vanishing Polyakov loops of non-trivial N-ality. A practical method of constraining to this orbit space C0 is found by implementing a certain axial-type gauge. It is shown that the representative of an orbit in C0 is unique in this gauge up to time independent Abelian gauge transformations. The restricted orbit space does not admit non-Abelian monopoles. As long as C0 is thermodynamically stable, the free energy of the constrained SU(N) gauge model is of order N0 (even in the presence of dynamical quarks) and confinement is manifest for sufficiently large N. With a free energy of order N0 and Polyakov loops that vanish by design, there is no transition that deconfines color charge in such an SU(N) model. However, a proliferation of massless hadronic states of arbitrary spin could lead to a Hagedorn transition[1] if the string tension vanishes at a finite temperature TH. Constraining the orbit space to C0 can be viewed as a particular boundary condition, and TH in general is above the first order deconfinement transition of the full theory at Td. Between Td and TH a superheated confining phase may exist for SU(N > 2). Perturbation theory in C0 is sketched. It does not suffer from the severe IR-divergences observed by Linde[2]for the ordinary high temperature expansion. Correlations of the lowest transverse Abelian Matsubara modes develop a renormalization group invariant pole of second order at vanishing spatial momentum transfer when T = TH. The latter could be associated with linear confinement.