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Let $(M,g)$ be a riemannian manifold with connections $\nabla^t$ of Ricci curvature $Ric(\nabla^t)$. I define a Ricci flow of connections:

$$g(\frac{\partial \nabla_X}{\partial t} Y,Z)=X.Ric(\nabla)(Y,Z)-Ric(\nabla)(\nabla_X Y,Z)-Ric(\nabla)(Y,\nabla_X Z)$$

Have we solutions for the Ricci flow of connections for short time?

We may also take:

$$\frac{\partial \nabla_X}{\partial t}Y= \nabla_X Ric(\nabla)Y-Ric(\nabla)\nabla_X Y$$

when $Ric$ is viewed as an endomorphism.

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