The double connections

Question

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

Answer 1

Writing your question in a more familiar language for me: Given

$D_X  Y=X^\mu\partial_\mu Y^\nu+A^{\nu\mu\rho}X_\mu Y_\rho$

$D'_X Y=X^\mu\partial_\mu Y^\nu+B^{\nu\mu\rho}X_\mu Y_\rho$

$A^{\nu\mu\rho}=A^{\nu\rho\mu}$

$B^{\nu\mu\rho}=B^{\nu\rho\mu}$

$\partial^\mu g_{\nu\rho}=A^{\nu\mu\rho}+B^{\nu\mu\rho}$

$\partial^\mu w_{\nu\rho}=A^{\tau\mu\nu} w_\tau\,^\rho+B^{\tau\mu\rho} w^\nu\,_\tau$

Can we reflect the variables $A$ and $B$? Because if we can, calculating the curvature is almost trivial. At least we know that $A^\mu\,_{\nu\rho}+B^\mu\,_{\nu\rho}=2\Gamma^\mu\,_{\nu\rho}$ (the Christoffel Symbols from the levi-civita conection). I'm not sure how to solve the last equation, now I don't have enough time to do that, sorry.

Comments

The first equation gives $A+B$ and the second equation $A-B$. If $dw=0$, then we have $A=B$, the Levi-Civita connection.