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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Qmechanic
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Conformal Transformations are defined as angle-preserving transition maps. $g(\mathbf{x}) \rightarrow \kappa^{2}(\mathbf{x}^{\prime}) g (\mathbf{x}^{\prime}) $. So in order, no and yes. – Gattu Mytraya Oct 12 '22 at 00:13
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Possible duplicates: https://physics.stackexchange.com/q/38138/2451 , https://physics.stackexchange.com/q/469205/2451 and links therein. – Qmechanic Oct 12 '22 at 06:28