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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.2012-11-05
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    @wj32 Isn't the second term a bit strange? The second condition doesn't preclude $f$ from being the zero function.2012-11-05
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    By "similarity" do you mean "isomorphism"? If $f=0$ then $Df(p)=0$ everywhere.2012-11-05
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    @wj32 By "similaritity", I mean [a composition of an isometry with a homothety](http://en.wikipedia.org/wiki/Similarity_%28geometry%29).2012-11-05
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    In that case it's fine unless your scaling sends everything to 0.2012-11-05
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    @wj32 Well, it's true that if a similarity isn't zero than it's an isomorphism. But not every isomorphism is a similarity. For example there is no real number $x$ such that $\begin{pmatrix}x&0\\0&x\end{pmatrix}\begin{pmatrix}0&1\\2&3\end{pmatrix}$ is an isometry.2012-11-05
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    Oh yes, you're right. Please ignore my second suggestion then.2012-11-05

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