Motivated by physics$^\dagger$, I am interested in manifolds which solve,
$$R_{ab} = -\alpha R_{acde} R^{cde}_b$$
where $\alpha >0$ and we have the usual Riemann and Ricci tensors. I know that for $R_{ab} = 0$ the fact the manifold is Ricci flat has implications on for example, its holonomy.
Are solutions to the above constraint known or studied in the mathematical literature?
$\dagger$ These are the Einstein equations with first order modifications by string theory.