Self-consistent compactification at finite temperature on R times S sup 5 times S sup 3

The self-consistency equations resulting from the Einstein equations for a space-time of the form {ital R}{times}{ital S}{sup 5}{times}{ital S}{sup 3}, with the vacuum-averaged energy-momentum tensor of a minimally coupled scalar field as the source, are solved using a one-loop finite-temperature calculation of this tensor. Solutions for low temperature are found to exist for large and small values of the radius ratio and also for the ratio close to 1/ {radical}2 . For the ratio equal to 1/ {radical}2 a zero-temperature solution is found. There is a maximum temperature for the ratio larger than this.