Some generalizations of the Riemann spaces of Einstein

We introduce a class of Riemann structures, called generalized Einstein structures of index $2e$, of which Einstein spaces are a particular case. We show that these structures are stationary for functions introduced on a family of Riemann structures of the compact manifold of H. Weyl. This result solves a problem of M. Berger. As examples of structures which are generalized Einstein structures over all indices we cite homogeneous compact Riemann spaces with a nondecomposable isotropy group and products of such spaces.