Continuous horizontally rigid functions of two variables are affine

Abstract

Cain, Clark and Rose defined a function ${{f\colon\mathbb{R}^n \to \mathbb{R}}}$ to be vertically rigid if graph(cf) is isometric to graph(f) for every c ≠ 0. It is horizontally rigid if graph${(f(c \vec{x}))}$ is isometric to graph(f) for every c ≠ 0 (see Cain et al., Real Anal. Exch. 31:515–518, 2005/2006). In Balka and Elekes (J. Math. Anal. Appl. 345:880–888, 2008) 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 ${{a+be^{kx} (a,b,k \in \mathbb{R})}}$. Later they proved in Balka and Elekes (Real. Anal. Exch. 35:139–156, 2009) 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 ${{a + be^{kx} + dy (a,b,d,k \in \mathbb{R}, k \neq 0, s\colon \mathbb{R}\to \mathbb{R}\,{\rm is\, continuous})}}$. The problem remained open in higher dimensions. The characterization in the case of horizontal rigidity is surprisingly simpler. Richter (Real Anal. Exch. 35:343–354, 2009) proved that a continuous function of one variable is horizontally rigid if and only if it is of the form ${{a+bx (a,b\in \mathbb{R})}}$. 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 ${{a+ bx + dy (a,b,d \in \mathbb{R})}}$. This problem also remains open in higher dimensions. The main new ingredient of the present paper is the use of functional equations.