OR:
A function \(f : \R^n \to \R^n\) is conformal if its Jacobian at any point is proportional to a rotation.
For \(n=2\), the Cauchy-Riemann equations make clear that conformal transformation are precisely (invertible) complex analytic maps.
or¶
A conformal transformation satisfies \(g_{\mu\nu}(y(x)) = \Omega(x)g_{\mu\nu}(x)\), where \(g\) on the left side is a function of the new coordinates \(y\). TODO: update ^
From just this constraint, we can derive a huge amount of information about the nature of conformal transformations, particularly in 2D.
For \(y^\mu = x^\mu + \epsilon^\mu\), with \(\epsilon\) small (i.e. considering only first order terms):
\[
g_{\mu\nu}(y) = \pd{x^\alpha}{y^\mu}\pd{x^\beta}{y^\nu} g_{\alpha\beta}(x) = (\pd{x^\alpha}{y^\mu}\pd{x^\beta}{y^\nu} - (\pd{\epsilon^\alpha}{y^\mu} + \pd{\epsilon^\beta}{y^\nu}))g_{\alpha\beta}(x)
\]
todo: what ^ ???
special conformal transformation as inversion, shift, inversion