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Instead of a plane, if we had an image on a sphere $x^2+y^2+z^2=1$ and wanted to conformally map this into another image on a sphere. What would the equivalent Cauchy-Riemann equations be for this?

i.e. we would have the coordinates of the new image $(x',y',z')$ such that:

$x' = f(x,y,z), y'=g(x,y,z), z'=h(x,y,z)$.

and $f^2+g^2+h^2=1$ and all angles preserved. I presume one could form some differential equation.

Bernard
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zooby
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  • I'd say for $\phi$ a differentiable map $S^2 \to S^2$ you need to define conformal, ie. a way to compare the angles at $a$ and $\phi(a)$, the obvious way is to say that every $M \in SO(3)$ is conformal, then $\phi$ is conformal iff $\forall a,v \in S^2\subset \Bbb{R}^3,\phi(v) = M_a (a+r_a(v-a))+o(|v-a|)$ for some $M_a \in SO(3), r_a > 0$ – reuns Apr 24 '19 at 21:19

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