New property of quaternionic Fueter functions (Q530122)

In this article dealing with quaternionic analysis, the author proves a rigidity result on quaternionic functions which are left-regular in the sense of Fueter, i.e., satisfy a property which is a quaternionic generalization of the Cauchy-Riemann equations in complex analysis. More precisely, let \({\mathbb H}\) denote the skew field of quaternions and let \(F:{\mathbb H}\rightarrow{\mathbb H}\) be a function which is left-regular in the sense of Fueter (for a definition see Equation (1) in the article). The main result in the paper is Theorem 2 which states that if \(q\in{\mathbb H}\) and the norm of \(F(q)\in{\mathbb H}\) can be written as \(| F (q)| = g(| q| )\) where \(g\) is a real-analytic function, then \(g\) is constant. From an analytic viewpoint the non-commutative field \({\mathbb H}\) is complete with respect to its standard Euclidean norm; hence it is a natural idea -- and also motivated by recent investigations in non-commutative geometry and physics -- to generalize the classical concepts of real or complex analysis such as differentiability, analyticity, integration, etc. for quaternionic functions. However apparently this is only a partially successful and technically difficult part of mathematics basically due to the non-commutativity of the quaternions. For example by the straightforward generalization of differentiability only linear functions are differentiable (see Theorem 1 in the article and the references therein); moreover there are several quaternionic generalizations of complex-valued holomorphic functions (one of them is the left- and right-regularity in the sense of Fueter) leading to inequivalent classes of ``quaternionic holomorphic'' functions and all of these function classes seem to be very restricted.