We introduce the notion of tame rho-quaternionic manifold that permits the construction of a finite family of p-connections, significant for the geometry involved. This provides, for example, the following: A new simple global characterisation of flat (complex-)quaternionic manifolds. A new simple construction of the metric and the corresponding Levi-Civita connection of a quaternionic-Kahler 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-Kahler manifolds are studied. (C) 2021 Elsevier B.V. All rights reserved.
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