## On stability of pseudo-isometries.(Russian)Zbl 0718.51009

Tr. Inst. Mat. 14, 89-98 (1989).
In the space $${\mathbb{R}}^ n$$ we define: $$\pi_ 1(x_ 1,...,x_ n)=(x_ 1,...,x_ k,0,...,0)$$, $$\pi_ 2(x_ 1,...,x_ n)=(0,...,0,x_{k+1},...,x_ n)$$, $$D(x)=| \pi_ 1(x)|^ 2- | \pi_ 2(x)|^ 2$$. A mapping $$\phi$$ : $${\mathbb{R}}^ n\to {\mathbb{R}}^ n$$ is called pseudo-isometry if for $$x,y\in {\mathbb{R}}^ n:$$ $${\mathcal D}(\phi (x)-\phi (y))={\mathcal D}(x-y)$$. Let U be a domain in $${\mathbb{R}}^ n$$, f: $$U\to {\mathbb{R}}^ n$$ a homeomorphism, and $$0\leq \epsilon <1$$. The mapping f is said to be a quasi-pseudo-isometry if for arbitrary $$\epsilon '$$ $$(\epsilon <\epsilon '<1)$$ and $$x_ 0\in U$$ there exists $$\delta >0$$ such that for every x, y from the ball $$B(x_ 0,\delta)$$ the following equality holds: $${\mathcal D}(f(x)-f(y))={\mathcal D}(x- y)+\theta (x,y)/(x-y)^ 2$$, where $$| \theta (x,y)| <\epsilon '$$. The set of all such mappings of U will be denoted by QPI($$\epsilon$$,U).
The main result of the paper is the following theorem: Let $$U\subset {\mathbb{R}}^ n$$ be a domain in $${\mathbb{R}}^ n$$, $$0\leq \epsilon <1$$ and $$f\in QPI(\epsilon,U)$$. Then there exist such constants $$\alpha >1$$, $$C>0$$, depending on n only, and a pseudo-isometry $$\phi$$ such that if U contains the ball $$B(x_ 0,\epsilon R)$$ then for $$x\in B(x_ 0,R)$$ the inequality $$| \phi [f(x)]-x| <C\epsilon R$$ holds.

### MSC:

 51F20 Congruence and orthogonality in metric geometry

pseudo-isometry