# Why is the absolute gradient of the metric tensor $\nabla_{\alpha} g_{\mu \nu} = 0$ in every coordinate system? [duplicate]

Is there any intuitive explanation for why the absolute gradient of the metric tensor $\nabla_{\alpha} g_{\mu \nu} = 0$ in every coordinate system?

## marked as duplicate by AccidentalFourierTransform, Ben Crowell, sammy gerbil, Kyle Kanos, Qmechanic♦Jun 14 '18 at 11:25

By the fundamental theorem of Riemannian geometry, on a manifold $M$ with metric $g$, it is always possible to choose a connection $\nabla$ such that,
$$\partial_X \langle Y,Z\rangle = \langle \nabla_X Y,Z\rangle + \langle Y, \nabla_X Z \rangle$$
where $X,Y$ and $Z$ are vector fields. Converting to explicit index notation, it is possible to show that this condition implies we can always choose a connection such that $\nabla_a g_{bc} = 0$.
It should be stressed, this choice of connection, the Levi-Civita connection (which has the added condition of being torsion free) is only one choice of connection, for the tangent bundle on $M$, and there are of course other choices and other bundles to consider, for which it is not true.