The flow of curvature

Question

Let $(M,g)$ be a riemannian surface with a family of metric connections $\nabla$. As the connection is metric, the curvature $R(X,Y)$ is antisymmetric so that we have proper values $\omega_i$ and vectors $e_i$: 

$$R(X,Y)e_k = i \omega_k (X,Y) e_k$$

with $\omega_k \in \Lambda^2 (TM)$. The proper values can be viewed as antisymmetric endomorphisms by the metric $\Omega_k \in End(TM)$. A flow of curvature can be defined by the formula:

$$\frac{\partial g}{\partial t}(X,Y)= -\sum_k g(\Omega_k X, \Omega_k Y)$$

Can we have solutions of the flow for small times ?