Let (M,g,w) be a manifold with a metric and a non-degenerated 2-form. I define two connections by the following formulas: X.g(Y,Z)=g(∇XY,Z)+g(Y,∇′XZ)
X.w(Y,Z)=w(∇XY,Z)+w(Y,∇′XZ)
And the connections are torsion free. Can we define the Ricci curvatures ?