Talk:Metric connection

merge metric compatibility here?

metric compatibility seems to be the exact same mathematical concept, just applied in the more restricted context of GR. Hence it should either be wholly integrated into this article, or (more simply for now) moved to be a subsection of this article. There's no need to split efforts among two articles, and that one is even more stubbish presently than this. Cesiumfrog (talk) 23:36, 2 September 2011 (UTC)

• Done. -- P 1 9 9  ✉ 17:04, 29 October 2014 (UTC)

Riemannian connection and torsion

Metric connection#Riemannian connection is confusing because $\nabla _{X}Y-\nabla _{Y}X=[X,Y]$ is equivalent to $T(X,Y)=0$, i.e., vanishing torsion.

I've removed this claim, since it conflicts with the immediately preceding definition

This is a connection $\nabla$ on the tangent bundle of a pseudo-Riemannian manifold (M, g) such that $\nabla _{X}g=0$ for all vector fields X on M. Equivalently, $\nabla$ is Riemannian if the parallel transport it defines preserves the metric g.

The constraint of being torsion-free is usually added to what is termed the Riemannian connection to define a Levi-Civita connection. If it was part of the definition of a Riemannian connection, such a connection would already be a Levi-Civita connection. The few notable sources that I have looked at are consistent with this view. —Quondum 17:27, 16 April 2021 (UTC)