  We show that all spatial gluon connected correlation functions in SU(N) or
SO(N) QCD vanish at finite temperature and zero momentum in lattice Landau or
Coulomb gauges, due to the proximity of the Gribov horizon. These observations
also apply to QCD with two colors and an even number of flavors at large
chemical potential. These nonperturbative results may have consequences on the
nature of the thermal magnetic mass and the character of the magnetic color
superconductivity.
