Harmonic polynomial morphisms between Euclidean spaces
It is an interesting problem to discuss the existence and nonexistence of harmonic polynomial morphisms between Euclidean spaces. The author shows that a nonsingular multilinear map \(f:\mathbb{R}^{p_1}\times \cdots\times \mathbb{R}^{p_k}\to \mathbb{R}^n\) is a harmonic morphism if and only if \(p_1 = p_2 =\dots = p_k\), \(n=1,2,4,8\). Moreover, by Cartan-Münzner's differential equations and a result of Baird, for \(p\geq 3\), the author obtains an isoparametric polynomial of degree \(p\), \(f : \mathbb{R}^N\to\mathbb{R}\) with \(m_1 = M - 2\) if there exists no orthogonal matrix \(A\) such that \(\langle \nabla g_x,\nabla f_x\rangle=0\) for all \(x\in\mathbb{R}^n\) where \(g = fA'\). Related papers are [\textit{P. Baird}, Harmonic maps with symmetry, harmonic morphisms and deformations of metrics, Research Notes in Mathematics 87 (1983; Zbl 0515.58010), \textit{T. Ishihara}, J. Math. Kyoto Univ. 19, 215-229 (1979; Zbl 0421.31006), and \textit{É. Cartan}, Math. Z. 45, 335-367 (1939; Zbl 0021.15603)].
scientific article; zbMATH DE number 1531237
scientific article