We prove an existence result for the prescribed Ricci curvature equation for certain doubly warped product metrics on $mathbb{S}^{d_1+1}times mathbb{S}^{d_2}$, where $d_i geq 2$. If $T$ is a metric satisfying certain curvature assumptions, we show th
at $T$ can be scaled independently on the two factors so as to itself be the Ricci tensor of some metric.
We show that a basis of a semisimple Lie algebra for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being ``nice. Namely, the bracket of any two basis elements is a multiple of
another basis element. This extends the work of Lauret and Will cite{lw13} on nilpotent Lie algebras. We also give a characterization for diagonalizing the Ricci tensor for homogeneous spaces, and study the Ricci flow behavior of diagonal metrics on cohomogeneity one manifolds.
We exhibit the first examples of closed 4-manifolds with nonnegative sectional curvature that lose this property when evolved via Ricci flow.
