v 2 1 9 O ct 1 99 7 Cosmological Einstein - Yang - Mills equations ∗ †

We use a systematic construction method for invariant connections on homogeneous spaces to find the Einstein-SU (n)-Yang-Mills equations for Friedmann-Robertson-Walker and locally rotationally symmetric homogeneous cosmologies. These connections depend on the choice of a homomorphism from the isotropy group into the gauge group. We consider here the cases of the gauge group SU (n) and SO(n) where these homo-morphisms correspond to unitary or orthogonal representations of the isotropy group. For some of the simpler cases the full system of the evolution equations are derived, for others we only determine the number of dynamical variables that remain after some mild fixing of the gauge.