On a continuation of quaternionic and octonionic logarithm along curves and the winding number

TL;DR

This paper investigates the extension of the hypercomplex logarithm in quaternionic and octonionic algebras along paths, introducing a logarithmic manifold to address the challenges of defining continuous branches.

Contribution

The authors introduce the logarithmic manifold _\u2113^+ and analyze path lifts, providing a new geometric framework for hypercomplex logarithm continuation.

Findings

The logarithmic manifold __^+ is a diffeomorphism with __^+ and __^+ is an immersion.

Path lifting to __^+ is generally not guaranteed, even in simply connected domains.

The problem of extending the hypercomplex logarithm is equivalent to lifting paths to the logarithmic manifold.

Abstract

This paper focuses on the problem of finding a continuous extension of the hypercomplex logarithm along a path. While a branch of the complex logarithm can be defined in a small open neighbourhood of a strictly negative real point, no continuous branch of the hypercomplex logarithm can be defined in any open set $A\subset \mathbb K\setminus \{0\}$ which contains a strictly negative real point $x_0$ (here $\mathbb K$ represents the algebra of quaternions or octonions). To overcome these difficulties, we introduced the logarithmic manifold $\mathscr E_\mathbb K^+$ and then showed that if $q\in\mathbb K,\ q=x+Iy$ then $E(x+Iy) %= (\exp (x + Iy), Iy) = (\exp x \cos y + I\exp x \sin y, Iy)$ is an immersion and a diffeomorphism between $\mathbb K$ and $\mathscr E_\mathbb K^+$. In this paper, we consider lifts of paths in $\mathbb K\setminus\{0\}$ to the logarithmic manifold $\mathscr{E}^+_\mathbb K$; even though $\mathbb K \setminus \{0\}$ is simply connected, in general, given a path in $\mathbb K \setminus \{0\}$, the existence of a lift of this path to $\mathscr{E}^+_\mathbb K$ is not guaranteed. There is an obvious equivalence between the problem of lifting a path in $\mathbb K \setminus \{0\}$ and the one of finding a continuation of the hypercomplex logarithm $\log_{\mathbb K}$ along this path.