Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds

• Main Authors: Aliya Naaz Siddiqui, Bang-Yen Chen, Oguzhan Bahadir
• Format: Article
• Language: English
• Published: MDPI AG 2019-09-01
• Series: Mathematics
• Subjects: statistical warped product submanifold; statistical manifold; B.Y.Chen inequality; Casorati curvatures; statistical soliton
• Online Access: https://www.mdpi.com/2227-7390/7/9/797

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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="double-struck">R</mi> <msub> <mo>&#215;</mo> <mi mathvariant="fraktur">f</mi> </msub> <msub> <mi>N</mi> <mn>2</mn> </msub> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mn>1</mn> </msub> <msub> <mo>&#215;</mo> <mi mathvariant="fraktur">f</mi> </msub> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula>. 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&#8217;s inequality involving scalar curvature, the squared mean curvature, and the Laplacian of warping function (with respect to the Levi&#8722;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.