Let (M,g) be a riemannian manifold with riemannian curvature R.
rg(x,y,z,t)=g(R(x,y)z,t)
The Riemann flow for the metric g is defined by the following equation:
∂∂t[g(x,x)g(y,y)−g(x,y)2]=rg(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?