Algebraic Regularity over Quaternions and Regular Four-Manifolds

TL;DR

This paper introduces algebraic regular functions over quaternions, forming a real algebra that respects composition, and uses them to define a new class of four-manifolds called regular four-manifolds.

Contribution

It generalizes the Cauchy-Riemann system to quaternionic functions and constructs regular four-manifolds using these functions as transition maps.

Findings

Algebraic regular functions form a real associative algebra.

These functions respect composition.

Regular four-manifolds are constructed using algebraic regular functions.

Abstract

Based on a new generalization of Cauchy-Riemann system presented in this paper, we introduce a class of quaternion-valued functions of a quaternionic variable, which are called algebraic regular functions. The set of algebraic regular functions is not only a real associative algebra, but also respect the composition of functions. Using algebraic regular functions as transition maps, we introduce a class of four-manifolds called the regular four-manifolds.