The Structure of Continuous Rigid Functions of Two Variables

Abstract

A function $f : \mathbb{R}^n \to \mathbb{R}$ is called vertically rigid if $graph(cf)$ is isometric to $graph (f)$ for all $c \neq 0$. In [1] settled Janković's conjecture by showing that a continuous function $f : \mathbb{R}\to \mathbb{R}$ is vertically rigid if and only if it is of the form $a+bx$ or $a+be^{kx}$ ($a,b,k \in )$ prove that a continuous function $f :\mathbb{R}^2 \to \mathbb{R}$ is vertically rigid if and only if, after a suitable rotation around the $z$-axis, $f(x,y)$ is of the form $a + bx + dy$, $a + s(y)e^{kx}$ or $a + be^{kx} + dy$ ($a,b,d,k \in \mathbb{R}$, $k \neq 0$, $s : \mathbb{R} \to \mathbb{R}$ continuous). The problem remains open in higher dimensions.