Let $(M,g)$ be a riemannian manifold with riemannian curvature $R$.
$$r_g(x,y,z,t)=g(R(x,y)z,t)$$
The Riemann flow for the metric $g$ is defined by the following equation:
$$\frac{\partial}{\partial t}[g(x,x)g(y,y)-g(x,y)^2]=r_g(x,y,x,y)$$
The definition is coherent because if $x=y$, the result is zero.
Is the Riemann flow really well defined and has solutions?