I'm new to manifolds. I'm reading "An Intro to Manifolds" by Tu, when I can. A diffeomorphism is a conformal map if the pullback of the metric results in a conformally equivalent metric.
Are there any references about diffeomorphisms that permute angles (not conformal)? (If this is impossible, please help me understand why?)
A diffeomorphism in $\Bbb R^2,$ from a surface to a surface, $f:(0,1)^2\to (0,1)^2$ that permutes angles should take the original set of angles and reorder them so that an arbitrary angle $\theta$ between two curves $\gamma_1$ and $\gamma_2$ with intersection at point $p,$ is now located at a new point $p'\ne p.$
Notice below how the four angles are the same when considered as sets but have been permuted/re-arranged.
Like I said I'm not sure if this notion is possible.