*Article* **Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds**

**Aliya Naaz Siddiqui 1, Bang-Yen Chen 2,\* and Oguzhan Bahadir 3**


Received: 25 July 2019; Accepted: 26 August 2019; Published: 1 September 2019

**Abstract:** Warped products play crucial roles in differential geometry, as well as in mathematical physics, especially in general relativity. In this article, first we define and study statistical solitons on Ricci-symmetric statistical warped products R <sup>×</sup>f *N*2 and *N*1 <sup>×</sup>f R. Second, we study statistical warped products as submanifolds of statistical manifolds. For statistical warped products statistically immersed in a statistical manifold of constant curvature, we prove Chen's inequality involving scalar curvature, the squared mean curvature, and the Laplacian of warping function (with respect to the Levi–Civita connection). At the end, we establish a relationship between the scalar curvature and the Casorati curvatures in terms of the Laplacian of the warping function for statistical warped product submanifolds in the same ambient space.

**Keywords:** statistical warped product submanifold; statistical manifold; B.Y.Chen inequality; Casorati curvatures; statistical soliton

**MSC:** 53B30; 53C15; 53C25
