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Let's say we have a diffeomorphism. We know that under certain circumstances (invariance of the metric) this diffeomorphism is an isometry. Here is the part I'm not sure about. Is an isometry always just a statement about the principle of covariance? I.e., under an isometry, do the laws of physicsalwayslook identical in any coordinate system? Or is the isometry a necessary, but not sufficient condition, and there are more restrictions on the diffeomorphism?

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# Is an isometry always a statement about the principle of covariance?

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