# On tame $\rho$-quaternionic manifolds

## TL;DR

This paper introduces tame $
ho$-quaternionic manifolds, providing new characterizations, constructions, and generalizations for quaternionic and quaternionic-K"ahler manifolds using $
ho$-connections and twistor space methods.

## Contribution

It presents the concept of tame $
ho$-quaternionic manifolds, offering new ways to characterize, construct, and generalize quaternionic geometries and their twistor spaces.

## Key findings

- New global characterization of flat quaternionic manifolds
- Construction of metrics and Levi-Civita connections from twistor spaces
- Analysis of twistorial harmonic morphisms with one-dimensional fibres

## Abstract

We introduce the notion of tame $\rho$-quaternionic manifold that permits the construction of a finite family of $\rho$-connections, significant for the geometry involved. This provides, for example, the following: (1) a new simple global characterisation of flat (complex-)quaternionic manifolds, and (2) a new simple construction of the metric and the corresponding Levi-Civita connection of a quaternionic-K\"ahler manifold by starting from its twistor space; moreover, our method provides a natural generalization of this correspondence. Also, a new construction of quaternionic manifolds is obtained, and the properties of twistorial harmonic morphisms with one-dimensional fibres from quaternionic-K\"ahler manifolds are studied.