Harmonic maps into Lie groups and homogeneous spaces
\textit{F. C. Park} and \textit{R. W. Brockett} [Proc. 28th IEEE Conf. Decision Control (Tampa, FL, 1989) (IEEE, New York), 206-210 (1989; MR 91a:70005)] introduced the notion of harmonic maps into the robotics literature by studying maps \(f:\mathbb{R}^n\to SO(3)\) given by \(f(x_1,\ldots,x_n)=M\exp(x_1A_1)\cdot\ldots\cdot\exp(x_1A_n)\) where \(M,A_i\in SO(3)\). This was done to find global measures for the mobility of a mechanism. But in their paper they considered only spherical mechanisms and raised the question how harmonic maps of the type \(f:\mathbb{R}^n\to SO(3)\) could be described so to extend their results to spatial mechanisms. The authors of the paper extend a result of \textit{A. I. Pluzhnikov} [Sov. Math., Dokl. 27, 246-248 (1983; Zbl 0531.58019)] and obtain a characterization of any harmonic map into an arbitrary homogeneous space with an invariant Riemannian metric. They find harmonic maps into all three-dimensional non-compact Lie-Groups and specializing furthermore to \(SO(3)\) answer the question of Park and Brockett.
scientific article; zbMATH DE number 1034097
scientific article