A certain class of Einstein-Yang-Mills--systems

A class of $G$-invariant Einstein-Yang-Mills (EYM) systems with cosmological constant on homogeneous spaces $G / H$, where $G$ is a semisimple compact Lie group, is presented. These EYM--systems can be obtained in terms of dimensional reduction of pure gravity. If $G / H$ is a symmetric space, the EYM--system on $G / H$ provides a static solution of the EYM--equations on spacetime ${\Bbb R} \times G / H$. This way, in particular, a solution for an arbitrary Lie group $F$, considered as a symmetric space, is obtained. This solution is discussed in detail for the case $F = SU(2)$. A known analytical EYM--system on ${\Bbb R} \times S^3$ is recovered and it is shown - using a relation to the BPST instanton - that this solution is of sphaleron type. Finally, a relation to the distance of Bures and to parallel transport along mixed states is shown.