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I am getting confused if the conformal transformations are transformations of charts on the manifold, or something else. Basically, I can imagine a Euclidean plane, and two observers one with chart $x$ of rectangular grid, other with dilated grid $x' = 2x$, is this a dilatation, a part of conformal transformations? I mean does the manifold, in general cases as well, the same, i.e. $(M, g)$ and we are just talking about transformations between charts essentially?

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