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Statistical Manifolds with almost Quaternionic Structures and Quaternionic Kähler-like Statistical Submersions

In this paper we investigate statistical manifolds with almost quaternionic structures. We define the concept of quaternionic Kahler-like statistical manifold and derive the main properties of quaternionic Kahler-like statistical submersions, extending in a new setting some previous results obtained by K. Takano concerning statistical manifolds endowed with almost complex and almost contact structures. Finally, we give a nontrivial example and propose some open problems in the field for further research.