In this paper, we discuss statistical manifolds with almost contact sturctures. We define a Sasaki-like statistical manifold. Moreover, we consider Sasaki-like statistical submersions, and we study Sasaki-like statistical submersion with the property that the curvature tensor with respect to the affine connection of the total space satisfies the condition (2.12).

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We consider Kähler-like statistical manifolds, whose curvature tensor field satisfies a natural condition. For their statistical submanifolds, we prove a Chen first inequality and a Chen inequality for the invariant δ(2, 2). MSC: 53C05; 53C40where ∇ 0 is the Levi-Civita connection on M.Similar definitions can be considered for semi-Riemannian manifolds (see also [5]).A statistical structure is said to be of constant curvature ∈ R [6] if:for any vector fields X, Y, Z. The same equation holds for R * (X, Y)Z.A statistical structure of null constant curvature is called a Hessian structure. In [2,7], K. Takano considered a (semi-)Riemannian manifold M, g with an almost complex structure J, endowed with another tensor field J * of type (1, 1) satisfying:for vector fields X and Y on M, g . Then, M, g, J is called an almost Hermite-like manifold. It is easy to see that J * * = J, J * 2 = −I and g JX, J * Y = g (X, Y) . If J is parallel with respect to ∇, then M, g, ∇, J is called a Kähler-like statistical manifold [7]. One also has: g

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