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I'm wondering if these objects have a name or are studied. No one around me knows, so I thought to ask here.

Let $\Phi:\mathbb{R}^d\rightarrow \mathbb{R}^d$ be $C^2$-diffeomoprhism, and fix $p \geq 0$, is there a name/known characterization of the diffeomorphisms satisfying $$ d(\Phi(x),y)-d(x,\Phi^{-1}(y)) \in o(d(x,y)^p)? $$

For example, and isometry solves this for $p=0$ (exactly), but what about in general?

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  • $\begingroup$ This is not clear to me why such a class of diffeomorphisms should be interesting. Any motivation supporting the definition of this class? $\endgroup$
    – Piotr Hajlasz
    Commented Dec 6, 2018 at 14:46
  • $\begingroup$ Yes it is near isometric $\endgroup$
    – ABIM
    Commented Dec 6, 2018 at 14:47
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    $\begingroup$ I am not convinced. The definition lacks certain homogeneity. If $\Phi$ satisfies it, then it is not clear that $\Phi_v=\Phi+v$ ($v$-constant vector) satisfies it. There are many conditions that are satisfied by isometries and not every such condition is forth studying. You need a deeper motivation than just the fact isometries satisfy it. $\endgroup$
    – Piotr Hajlasz
    Commented Dec 6, 2018 at 14:52
  • $\begingroup$ @PiotrHajlasz hmmm I took some time to think about that, and its true; very interesting point! $\endgroup$
    – ABIM
    Commented Dec 20, 2018 at 22:34

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