Let $(M,g_t)$ be a riemannian manifold with riemannian curvature $R(g)$ and scalar curvature $r(g)$, I define:
$$X_g=dr(g)^*$$
then I can define a flow over metrics:
$$\frac{\partial g}{\partial t}(X,Y)=R(g)(X,X_g,Y,X_g)$$
It is symmetric in $X,Y$.
Is this flow well defined? Have we singularities for the flow?