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Let $f:\Bbb R^2\to\Bbb R^2$ be a differentiable function. Are there names for the following two conditions?

  1. $Df(p)$ is an isometry at each point $p\in\Bbb R^2$;

  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.

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I think they are called local isometries and local diffeomorphisms. – wj32 Nov 5 '12 at 2:08
@wj32 Isn't the second term a bit strange? The second condition doesn't preclude $f$ from being the zero function. – Bartek Nov 5 '12 at 2:35
By "similarity" do you mean "isomorphism"? If $f=0$ then $Df(p)=0$ everywhere. – wj32 Nov 5 '12 at 2:36
@wj32 By "similaritity", I mean a composition of an isometry with a homothety. – Bartek Nov 5 '12 at 2:39
In that case it's fine unless your scaling sends everything to 0. – wj32 Nov 5 '12 at 2:48

For the first one it is just that - an isometry (that's the same as local isometry if dimensions of target and domain agree). For the second one is related. A linear map that preserves angles will be a composition of homothety and an isometry. The coefficient of homothety is the "conformal factor". Being just similarity at each point probably does not have a name because it does not make sense on general manifolds.

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