The structure of continuous rigid functions of two variables

TL;DR

This paper characterizes continuous vertically rigid functions of two variables, showing they are affine, exponential, or affine plus exponential after rotation, extending prior one-dimensional results.

Contribution

It extends the classification of continuous vertically rigid functions from one dimension to two dimensions, identifying their explicit forms after rotation.

Findings

Continuous vertically rigid functions in two variables are affine, exponential, or affine plus exponential after rotation.

The classification in higher dimensions remains an open problem.

Abstract

A function $f:\RR^n \to \RR$ is called \emph{vertically rigid} if $graph(cf)$ is isometric to $graph (f)$ for all $c \neq 0$. We settled Jankovi\'c's conjecture in a separate paper by showing that a continuous function $f:\RR \to \RR$ is vertically rigid if and only if it is of the form $a+bx$ or $a+be^{kx}$ ($a,b,k \in \RR$). Now we prove that a continuous function $f:\RR^2 \to \RR$ 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 $a + be^{kx} + dy$ ($a,b,d,k \in \RR$, $k \neq 0$, $s : \RR \to \RR$ continuous). The problem remains open in higher dimensions.