Slice conformality and Riemann manifolds on quaternions and octonions (Q2172500)

Let \(A\) be either the real algebra of quaternions \(\mathbb{H}\) or the real algebra of octonions \(\mathbb{O}\). This article addresses an interesting problem: finding analogs over \(A\) of the Riemann surfaces over \(\mathbb{C}\). While Riemann's approach was based on the use of conformal maps, this is not a possible approach over \(A\). Indeed, a famous result due to Liouville states that in real dimension larger than \(2\) the only conformal maps are compositions of translations, similarities, orthogonal transformations and inversions. The authors of the present article therefore define a new and more flexible notion: the notion of \textit{slice conformal} or \textit{slice isothermal map}. This notion is compatible with the notion of slice regular function over \(A\), see [\textit{G. Gentili} et al., Regular functions of a quaternionic variable. Berlin: Springer (2013; Zbl 1269.30001)]. The authors of the present article apply their innovative approach to the quaternionic and octonionic Riemann spheres and to construct the quaternionic and octonionic helicoidal and catenoidal manifolds. Moreover, they construct the logarithm manifold over \(A\), which sets the ground to define local branches of the quaternionic or octonionic logarithm. For similar purposes, the authors conclude the article with the construction of the nth-root manifold.