I am trying to find a simple formula for Reimannian and Ricci curvature of a linear connection, but I derive complicated formula. My problem is as follows
A Reimannian manifold $(M^n , g)$ with Levi-Civita connection $\nabla$ is given. I want to compute the $\bar{R}$ curvature of the following linear connection $$\overline{\nabla}_X Y =\nabla_X Y + {1\over6}N(X,Y)+ {1\over6}N(Y,X)$$ where, $N$ is a $(2,1)$-tensor field. Then, what will be the Ricci tensor $$\overline{Ric}(X,Y)=tr_g (Z\to \bar{R}(X,Z)(Y))$$?
Any comment is appreciated.