Self-consistent Kaluza-Klein structures

The self-consistent Einstein equations for a static space-time with spatial section $S^3\times S^3$ are solved using the finite-temperature, one-loop energy-momentum tensor of a massless Dirac field as source. Three types of solution occur. Those of the form $S^3\times R^3$ are possible only if the temperature $T$ is less than some critical value $T_c$. Symmetrical solutions, with equal radii, exist for all $T$, while for $T$ less than some value (bigger than $T_c$) a nonsymmetrical solution, with finite radii, is allowed.