Can someone help me out with this idea? 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 physics always look identical in any coordinate system? Or is the isometry a necessary, but not sufficient condition, and there are more restrictions on the diffeomorphism?