# The double connections

+ 1 like - 0 dislike
147 views

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 ?

edited May 19, 2020

+ 0 like - 0 dislike

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.

answered May 19, 2020 by (35 points)

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

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOve$\varnothing$flowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.