Let (M,g) be a riemannian manifold with connections ∇t of Ricci curvature Ric(∇t). I define a Ricci flow of connections:
g(∂∇X∂tY,Z)=X.Ric(∇)(Y,Z)−Ric(∇)(∇XY,Z)−Ric(∇)(Y,∇XZ)
Have we solutions for the Ricci flow of connections for short time?
We may also take:
∂∇X∂tY=∇XRic(∇)Y−Ric(∇)∇XY
when Ric is viewed as an endomorphism.