**7. Examples**

We provide examples of statistical warped product submanifolds as follows:

**Example 1.** *By generalizing Example 2.7 of [10] to higher dimensions, we see that:*

$$\left(\mathbb{R}\times\_{\mathfrak{e}^z}\mathbb{R}^n,\,\widetilde{\mathfrak{g}}=\,dz^2+\mathfrak{e}^{2z}(d\mathfrak{x}\_1^2+\cdots+d\mathfrak{x}\_n^2),\,\nabla\_\prime\nabla^\*\right)$$

*is a statistical warped product manifold. Furthermore, the hyperbolic space:*

$$\mathbb{H}^{n+1}(-1) = \left( \left\{ (\mathbf{x}\_0, \dots, \mathbf{x}\_{n+1}) \in \mathbb{R}^{n+1} | \mathbf{x}\_0 > 0 \right\}, \mathfrak{g} = \frac{d\mathbf{x}\_0^2 + \dots + d\mathbf{x}\_{n+1}^2}{x\_0^2}, \tilde{\nabla}\_\mathbf{v} \tilde{\nabla}^\* \right)$$

*is the statistical manifold of constant sectional curvature* <sup>−</sup>1*. Thus, with respect to the Levi–Civita connection,* R ×*e<sup>z</sup>* R*<sup>n</sup>*−<sup>1</sup> *admits an isometric minimal immersion into* <sup>H</sup>*<sup>n</sup>*+<sup>1</sup>(−<sup>1</sup>)*.*

**Example 2.** (R ×*z* R*<sup>n</sup>*, *g*˜ = *dt*<sup>2</sup> + *t*<sup>2</sup>(*dx*21 + ··· + *dx*2*n*), ∇, ∇∗) *is a statistical warped product manifold, and it is isometric to the Euclidean* (*n* + <sup>1</sup>)*-space* E*<sup>n</sup>*+1*. Let N be a minimal submanifold of the unit hypersphere Sn*(1) ⊂ E*n*+<sup>1</sup> *center at the origin o* ∈ <sup>E</sup>*<sup>n</sup>*+1*, and let C*(*N*) *be the cone over N with the vertex at o.*

*The metric of C*(*N*) *is the warped product metric gC*(*N*) = *dt*<sup>2</sup> + *<sup>t</sup>*<sup>2</sup>*gN, where gN denotes the metric of N. Any open submanifold M of C*(*N*) *is a warped product manifold, which admits an isometric minimal immersion into the statistical manifold* E*n*+<sup>1</sup> *of constant sectional curvature zero.*

**Author Contributions:** Conceptualization, A.N.S., B.-Y.C. and O.B.; Methodology, A.N.S., B.-Y.C. and O.B.; Software, A.N.S. and B.-Y.C.; Validation, A.N.S., B.-Y.C. and O.B.; Formal Analysis, A.N.S.; Investigation, A.N.S., B.-Y.C. and O.B.; Resources, A.N.S.; Data curation, A.N.S. and B.-Y.C.; Writing-Original Draft Preparation, A.N.S.; Writing-Review and Editing, A.N.S. and B.-Y.C.; Visualization, A.N.S.; Supervision, B.-Y.C.; Project Administration, A.N.S., B.-Y.C. and O.B.; Funding Acquisition, A.N.S., B.-Y.C. and O.B.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors thank the referees for many valuable and helpful suggestions to improve the presentation of this article.

**Conflicts of Interest:** The authors declare no conflict of interest.
