Let $f:\Bbb R^2\to\Bbb R^2$ be a differentiable function. Are there names for the following two conditions?
$Df(p)$ is an isometry at each point $p\in\Bbb R^2$;
$Df(p)$ is a similarity at each point $p\in\Bbb R^2?$
(I'm interested in $\Bbb R^2$ mainly, but if there's a general term for all finite dimensions, then please let me know.)
I would like to know this because I noticed that if $g$ satisfies 2. and $f:\Bbb C\to\Bbb C$ is holomorphic, then $g^{-1}\circ f\circ g$ is holomorphic because a conjugation of a rotation by an isometry is a rotation, and scalings commute with everything so the composition's derivative is a scaled rotation, which makes the composition holomorphic.