Continuous horizontally rigid functions of two variables are affine

Abstract

Cain, Clark and Rose defined a function $f\colon \RR^n \to \RR$ to be \emph{vertically rigid} if $\graph(cf)$ is isometric to $\graph (f)$ for every $c\ne 0$. It is \emph{horizontally rigid} if $\graph(f(c \vec{x}))$ is isometric to $\graph (f)$ for every $c\ne 0$ (see \cite{CCR}). In an earlier paper the authors of the present paper settled Janković's conjecture by showing that a continuous function of one variable is vertically rigid if and only if it is of the form $a+bx$ or $a+b{e}^{kx}$ ($a,b,k \in \RR$). Later they proved that a continuous function of two variables is vertically rigid if and only if after a suitable rotation around the z-axis it is of the form $a+bx+dy$, $a+s(y)e^{kx}$ or $a+b{e}^{kx}+dy$ ($a,b,d,k \in \RR$, $k\ne 0$, $s : \RR \to \RR$ continuous). The problem remained open in higher dimensions. The characterization in the case of horizontal rigidity is surprisingly simpler. C. Richter proved that a continuous function of one variable is horizontally rigid if and only if it is of the form $a+bx$ ($a,b\in \RR$). The goal of the present paper is to prove that a continuous function of two variables is horizontally rigid if and only if it is of the form $a+bx+dy$ ($a,b,d \in \RR$). This problem also remains open in higher dimensions. The main new ingredient of the present paper is the use of functional equations.