A direct approach to quaternionic manifolds

TL;DR

This paper introduces the concept of quaternionic regular manifolds, extending the notion of slice regular functions to define manifolds modeled on quaternionic spaces, and provides examples including those with affine structures.

Contribution

It defines quaternionic regular manifolds using slice regularity and presents significant classes of such manifolds, including those with affine structures.

Findings

Defined quaternionic regular manifolds as spaces locally modeled on ^n

Provided examples of quaternionic regular manifolds, including affine structures

Extended the theory of slice regular functions to manifold settings

Abstract

The recent definition of slice regular function of several quaternionic variables suggests a new notion of quaternionic manifold. We give the definition of quaternionic regular manifold, as a space locally modeled on $\mathbb{H}^n$, in a slice regular sense. We exhibit some significant classes of examples, including manifolds which carry a quaternionic affine structure.