A flow of connections

Question

Let $(M,g)$ be a riemannian manifold, I define a flow of affine connections $\nabla_t$:

$$\frac{\partial \nabla_X}{\partial t}Y=R(dr^*,X)Y$$

where $R$ is the curvature of $\nabla$, and $r$ is the scalar curvature of $\nabla$ which is the trace of the Ricci curvature of $\nabla$ (the trace of $X\rightarrow R(X,Y)Z$).

Have we solutions of this flow for short time?