Definition talk:Metric Compatible Connection

Mistake
What happened here is that I left out the notion of general covariant derivative (for personal reasons, I only had time for most basic notions). The current definition at hand is defined only for vector fields. However, this covariant derivative induces a covariant derivative of arbitrary tensor fields including scalars. And for scalars it reduces to exactly what you wrote. So what we really have to do here is to created a theorem page for existence of such a derivative, and then create a definition page for the more general covariant derivative. Referring to that derivative would then solve the issue.--Julius (talk) 21:44, 26 May 2023 (UTC)


 * OK, this will be correct if you extend the definition of $\nabla_X$ so that especially $\nabla_X \innerprod Y Z = \map X {\innerprod Y Z}$. I think, the simplest and best solution is just to write it $\map X {\innerprod Y Z}$, as it actually is. But OK to leave it until you add the extended definition of $\nabla_X$. --Usagiop (talk) 22:12, 26 May 2023 (UTC)
 * Is this Definition:Koszul Connection not the extended definition? I think, you just need to correct the link. --Usagiop (talk) 22:38, 26 May 2023 (UTC)


 * Way over my head I'm afraid. I'll let the experts hammer this out. --prime mover (talk) 21:46, 26 May 2023 (UTC)