Is there any known class of conformal transformations $\phi : M \to M$ of a riemannian/semi-riemanian manifold $(M,g)$ that have the property: $g$ is ricci-positive iff $\phi^* g$ ricci positive?

There trivial answer is when $\phi^* g = cg$ for a constant c. Do we know something more than that? It seems that there are some results for maps that preserve the ricci curvature at each point (e.g. http://www.sciencedirect.com/science/article/pii/S0021782407000839 )

If we allow for not-conformal transformation is there something more we can say?

This post imported from StackExchange MathOverflow at 2017-02-19 11:21 (UTC), posted by SE-user Zakk