Spontaneous compactification from the point of view of dimensional reduction of gauge fields

The connection between dimensional reduction and spontaneous compactification is studied. It is shown that for symmetric internal spaces the Higgs vacuum of the reduced theory corresponds to a multidimensional field configuration satisfying the condition of parallelizability, a condition that is stronger than the Einstein–Yang–Mills equations and leads to spontaneous compactification. Explicit expressions are given for the solutions of the spontaneous compactification equations in the case of the internal space $CP^m$.