Optimizations on Statistical Hypersurfaces with Casorati Curvatures

Abstract

In the present paper, we study Casorati curvatures for statistical hypersurfaces. We show that the normalized scalar curvature for any real hypersurface (i.e., statistical hypersurface) of a holomorphic statistical manifold of constant holomorphic sectional curvature k is bounded above by the generalized normalized δ−Casorati curvatures and also consider the equality case of the inequality. Some immediate applications are discussed.

The Translation Surfaces on Statistical Manifolds with Normal Distribution

“…In recent years, during literature reviews on statistical submanifolds, it has been observed that studies have focused on Chen inequalities ( [5], [6], [15], [53]),Wintgen inequalities ( [8], [25], [44]) and inequalities involving the normalized δ-Casorati curvatures ( [12], [16], [17], [26], [38], [54]).…”

Bounds for Statistical Curvatures of Submanifolds in Kenmotsu-like Statistical Manifolds

“…In this reason, the geometric study of Casorati curvatures for submanifolds is new and has many research problems. A couple of optimal Casorati inequalities had been obtained by many distinguished geometers in different ambient space forms (for example, [3,4]). Decu et al have built certain inequalities for statistical submanifolds of Kenmotsu statistical manifolds with constant φ-sectional curvature involving normalized δ-Casorati curvatures and scalar curvature in [5].…”

“…Guadalupe and Rodriguez generalized Wintgen's inequality to a real-space surface of arbitrary codimension in the form R m+2 (c), m ≥ 2. After that, Chen extended this inequality to surfaces in a 4-dimensional pseudo-Euclidean space E 4 2 with a neutral metric. In [8], DeSmet, Dillen, Verstraelen, and Vrancken found the DDVV conjecture (called the generalized Wintgen inequality in general) for an isometric immersion of a Riemannian manifold into a real space form.…”