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

Abstract

In this paper we investigate statistical manifolds with almost quaternionic structures. We define the concept of quaternionic Kähler-like statistical manifold and derive the main properties of quaternionic Kähler-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.

Conformal generic submersions

“…Watson defined an almost Hermitian submersion between almost Hermitian manifolds and he showed that the base manifold and each fiber have the same kind of structure as the total space, in most cases. We note that almost Hermitian submersions have been extended to the almost contact manifolds [13], locally conformal Kähler manifolds [25], quaternionic Kähler manifolds [22], paraquaternionic manifolds [11], [39] and statistical manifolds [40].…”

Hemi-slant ξ⊥-Riemannian submersions in contact geometry

“…After this kind of submersions were studied between manifolds endowed with differentiable structures. Many authors studied different geometric properties of the Riemannian submersions, anti-invariant submersion [18,30,33], semi-invariant submersion [4,31], paraquaternionic 3-submersion [37], statistical submersion [38], slant submersion [11,12,15,27,32], semi-slant submersion [16,25,26], conformal slant submersion [2,17] , conformal semi-slant submersion [1], bi-slant submersion [34] and Quasi bi-slant submersion [28].…”

Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature

“…(see, e.g., [27][28][29][30][31][32]). In particular, the differential geometry field is focused on topics such as submanifold theory of statistical manifolds [33], Hessian geometry [34], statistical submersions [35], complex manifold theory of statistical manifolds ( [29,36,37]), contact theory on statistical manifolds [38], and quaternionic theory on statistical manifolds [39]. For the above problems, Aydin et al obtained Chen-Ricci inequalities [40] and a generalized Wintgen inequality [41] for submanifolds in statistical manifolds of constant curvature.…”