A flow of Ricci curvature

Question

Let $(M,g)$ be a riemannian manifold. I define a flow of Ricci curvature by the following equation:

$$\frac{\partial g}{\partial t}(X,Y)= \sum_{i=1}^n g(\nabla_{e_i}(Ric)(X),\nabla_{e_i}(Ric)(Y))$$

$(e_i)$ is an orthonormal basis and $Ric$ is the Ricci curvature.

Have we solutions of the flow of Ricci curvature for short time?