Slice conformality and Riemann manifolds on quaternions and octonions

Abstract

In this paper we establish quaternionic and octonionic analogs of the classical Riemann surfaces. The construction of these manifolds has nice peculiarities and the scrutiny of Bernhard Riemann approach to Riemann surfaces, mainly based on conformality, leads to the definition of slice conformal or slice isothermal parameterization of quaternionic or octonionic Riemann manifolds. These new classes of manifolds include slice regular quaternionic and octonionic curves, graphs of slice regular functions, the $4$ and $8$ dimensional spheres, the helicoidal and catenoidal $4$ and $8$ dimensional manifolds. Using appropriate Riemann manifolds, we also give a unified definition of the quaternionic and octonionic logarithm and $n$-th root function.