# Aleksandrov problem for isometric mappings

Let $X$, $Y$ be two metric spaces, with respective distances $d_1$, $d _ { 2 }$ (cf. also Metric space). A mapping $f : X \rightarrow Y$ is defined to be an isometry if $d_{2} ( f ( x ) , f ( y ) ) = d _ { 1 } ( x , y )$ for all $x , y \in X$. A mapping $f : X \rightarrow Y$ is said to preserve the distance $r$ if for all $x , y \in X$ with $d _ { 1 } ( x , y ) = r$ one has $d _ { 2 } ( f ( x ) , f ( y ) ) = r$. A.D. Aleksandrov has posed the problem whether the existence of a single preserved distance for some mapping $f$ implies that $f$ is an isometry (cf. [a1]).

Even if $X$, $Y$ are normed vector spaces, the above problem is not easy to answer (note that, in this case, taking $r = 1$ is no loss of generality). For example, the following question has not been solved yet (as of 2000): Is a mapping $f$ from $\mathbf{R} ^ { 2 }$ to $\mathbf{R} ^ { 3 }$ preserving unit distance necessarily an isometry (cf. [a17])?

The discussion of Aleksandrov's problem under certain additional conditions on the given mapping preserving unit distance has led to several interesting and new problems (cf. [a14], [a15], [a16], [a18], [a17]).

The Aleksandrov problem has been solved for Euclidean spaces $X = Y = \mathbf{R} ^ { n }$: For $2 \leq n < \infty$, the answer is positive [a2], while for $n = 1 , \infty$, the answer is negative [a2], [a5], [a13]. In general normed vector spaces, the answer is positive for a mapping that is contractive, surjective and preserves the unit distance. The problem also has a positive solution when $X$ and $Y$ are strictly convex vector spaces, provided $f$ is a homeomorphism and the dimension of $X$ is greater than $2$ (cf. [a12], [a20]).

The Aleksandrov problem has also been solved in some special cases of mappings $f : X \rightarrow Y$ which preserve two distances with an integer ratio greater than $1$, with $X$ and $Y$ strictly convex vector spaces and the dimension of $X$ greater than $1$ (cf. [a3], [a16], [a17]). Furthermore, when $X$, $Y$ are Hilbert spaces and the dimension of $X$ is greater than $1$, a lot of work has been done (cf. [a21], [a22], [a23]; for example, when $f$ preserves the two distances $1$ and $\sqrt { 3 }$; when $\operatorname{dim} X \geq 3$ and $f$ preserves $1$ and $\sqrt { 2 }$, then $f$ is an affine isometry, etc.).

Problems connected with stability of isometries as well as non-linear perturbations of isometries have been extensively studied in [a4], [a5], [a6], [a7], [a8], [a9], [a10], [a11].

How to Cite This Entry:
Aleksandrov problem for isometric mappings. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Aleksandrov_problem_for_isometric_mappings&oldid=50017
This article was adapted from an original article by Shuhuang Xiang (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article