We study the Ricci iteration for homogeneous metrics on spheres and complex projective spaces. Such metrics can be described in terms of modifying the canonical metric on the fibers of a Hopf fibration. When the fibers of the Hopf fibration are circles or spheres of dimension 2 or 7, we observe that the Ricci iteration as well as all ancient Ricci iterations can be completely described using known results. The remaining and most challenging case is when the fibers are spheres of dimension 3. On the 3-sphere itself, using a result of Hamilton on the prescribed Ricci curvature equation, we establish existence and convergence of the Ricci iteration and confirm in this setting a conjecture on the relationship between ancient Ricci iterations and ancient solutions to the Ricci flow. In higher dimensions we obtain sufficient conditions for the solvability of the prescribed Ricci curvature equation as well as partial results on the behavior of the Ricci iteration.

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair (g,c) consisting of a left‐invariant Riemannian metric g and a positive constant c such that Ric(g)=cT, where Ric(g) is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that Ric(g)=cT is solvable for some left‐invariant Riemannian metric g.

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Let G/H be a compact homogeneous space, and letĝ 0 andĝ 1 be G-invariant Riemannian metrics on G/H. We consider the problem of finding a G-invariant Einstein metric g on the manifold G/H × [0, 1] subject to the constraint that g restricted to G/H × {0} and G/H × {1} coincides withĝ 0 andĝ 1 , respectively. By assuming that the isotropy representation of G/H consists of pairwise inequivalent irreducible summands, we show that we can always find such an Einstein metric.

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