Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Slice conformality and Riemann manifolds on quaternions and octonions

Graziano Gentili, Jasna Prezelj, Fabio Vlacci

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.

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 and dimensional spheres, the helicoidal and catenoidal and dimensional manifolds. Using appropriate Riemann manifolds, we also give a unified definition of the quaternionic and octonionic logarithm and -th root function.

Paper Structure

This paper contains 14 sections, 12 theorems, 116 equations.

Table of Contents

  1. Preface
  2. Preliminaries
  3. Parameterized quaternionic and octonionic Riemann manifolds
  4. The standard set of curves and the case of the differential of a slice regular function
  5. The differential of a smooth slice function
  6. Slice conformal curves
  7. Quaternionic and octonionic slice regular curves
  8. Other examples of hypercomplex Riemann manifolds
  9. The Riemann sphere
  10. The helicoidal hypercomplex manifold
  11. The catenoidal hypercomplex manifold
  12. The hypercomplex logarithm and $n$-th root
  13. The hypercomplex logarithm
  14. The hypercomplex $n$-th root

Key Result

Proposition 2.4

If a slice domain $\Omega$ in $\mathbb{K}$ is axially symmetric, then any slice regular function $f:\Omega \to \mathbb{K}$ is induced by a holomorphic stem function $F : D = \Omega_i \rightarrow \mathbb{K}_{\mathbb{C}}$.

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • proof
  • ...and 32 more