Let $(M,g)$ be a riemannian manifold with riemannian curvature $R$. We have $d^*_{\nabla} =* \circ d_{\nabla} \circ *$, with $*$ the Hodge operator, and $d_{\nabla}$ the differential so that we can define:

$$d^*_{\nabla} R \in \Lambda^1 (TM) \otimes End(TM)$$

We define a flow over the metrics called the Riemann flow by:

$$ \frac {\partial g}{\partial t}(X,Y)= -tr((d^*_{\nabla} R(X))\circ (d^*_{\nabla} R(Y)))$$

Has the Riemann flow solutions for any initial conditions?

Moreover, the Einstein-Riemann metrics are such that:

$$\lambda g(X,Y)= tr((d^*_{\nabla} R(X))\circ (d^*_{\nabla} R(Y)))$$

with $\lambda$ a scalar.