**1. Introduction**

Statistical manifolds were introduced in 1985 by S. Amari [1] in terms of information geometry, and they were applied by Lauritzen in [2]. Such manifolds have an important role in statistics as the statistical model often forms a geometrical manifold.

Let ∇ ˜ be an affine connection on a (pseudo-)Riemannian manifold (*N*˜ , *g*˜). The affine connection ∇ ˜ ∗ on *N* ˜ satisfying:

$$E\tilde{\mathcal{g}}(F,G) = \tilde{\mathcal{g}}(\tilde{\nabla}\_E F, G) + \tilde{\mathcal{g}}(F, \tilde{\nabla}\_E^\* G), \ \forall E, F, G \in \Gamma(T\tilde{N}),$$

is called a *dual connection* of ∇ ˜ with respect to *g*˜.

The triplet (*N*˜ , ∇˜ , *g*˜) is called a *statistical manifold* if:


If (∇˜ , *g*˜) is a statistical structure on *N*˜ , then (∇˜ ∗, *g*˜) is also a statistical structure. The connections ∇ ˜ and ∇ ˜ ∗ satisfy (∇˜ ∗)∗ = ∇˜ . On the other hand, we have ∇˜ 0 = 12 (∇˜ + ∇˜ <sup>∗</sup>), where ∇˜ 0 is the Levi–Civita connection of *N* ˜ .

One of the most fruitful generalizations of Riemannian products is the warped product defined in [3]. The notion of warped products plays very important roles in differential geometry and in mathematical physics, especially in general relativity. For instance, space-time models in general relativity are usually expressed in terms of warped products (cf., e.g., [4,5]).

In 2006, L. Todjihounde [6] defined a suitable dualistic structure on warped product manifolds. Furthermore, Furuhata et al. [7] defined Kenmotsu statistical manifolds and studied how to construct such structures on the warped product of a holomorphic statistical manifold [8] and a line. In [9], H. Aytimur and C. Ozgur studied Einstein statistical warped product manifolds. Further, C. Murathan and B. Sahin [10] studied and obtained the Wintgen-like inequality for statistical submanifolds of statistical warped product manifolds.

The Ricci solitons are special solutions of the Ricci flow of the Hamilton. In Section 4, we define statistical solitons and study the problem under what conditions the base manifold or fiber manifold of a statistical warped product manifold is a statistical soliton.

Curvature invariants play the most fundamental and natural roles in Riemannian geometry. A fundamental problem in the theory of Riemannian submanifolds is (cf. [11]):

**Problem A.***"Establish simple optimal relationships between the main intrinsic invariants and the main extrinsic invariants of a submanifold."*

The first solutions of this problem for warped product submanifolds were given in [11,12]. In Section 5, we study this fundamental problem for statistical warped product submanifolds in any statistical manifolds of constant curvature. Our solution to this problem given in this section is derived via the fundamental equations of statistical submanifolds.

An extrinsic curvature of a Riemannian submanifold was defined by Casorati in [13], as the normalized square of the length of the second fundamental form. Casorati curvature has nice applications in computer vision. It was preferred by Casorati over the traditional curvature since it corresponds better to the common intuition of curvature.

Several sharp inequalities between extrinsic and intrinsic curvatures for different submanifolds in real, complex, and quaternionic space forms endowed with various connections have been obtained (e.g., [14–21]). Such inequalities with a pair of conjugate affine connections involving the normalized scalar curvature of statistical submanifolds in different ambient spaces were obtained in [22–26].

Inspired by historical development on the classifications of Casorati curvatures and Ricci curvatures, we establish in Section 6 an inequality for statistical warped product submanifolds in a statistical manifold of constant curvature. In the last section, we provide two examples of statistical warped product submanifolds in the same environment.
