2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/206704We study eta-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem for Sasakian geometry to prove the existence of eta-Einstein structures on many different compact manifolds, including exotic spheres. We also relate these results to the existence of Einstein-Weyl structures.31 pages, minor changes made, to appear in Commun. Math. PhysDifferential GeometryHigh Energy Physics - Theorytext