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I am reading Sean Carroll's note on gr and he mentioned metric compatibility.

When ∇g=0 we say the metric is compatible.

However from another online material, the lecturer argues ∇ of a tensor is still a tensor,

and given that ∇g vanish in locally flat coordinate and this is a tensorial equation, therefore it vanishes in any other coordinate. That gives us ∇g always = 0.

I guess the contradiction comes from some implicit use about ∇g=0 vanish in locally flat coordinate but I am not sure what exactly is it. The first derivative in local coordinate vanishes, but I am not sure if the connection symbol vanishes too. I mean in locally flat coordinate the metric is cartesian like, but does that immediately imply the connection is also cartesian like (=0)?

Does anyone know why is there a contradiction? Sean Carroll and Schutz did not talk much about non-torsion-free cases so I really don't know what is going on.

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# I When will metric compatibility hold/not hold?

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