On tame rho-quaternionic manifolds

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.