Optimal inequalities for submanifolds in statistical manifolds of quasi constant curvature

Abstract

In this paper, we establish B.-Y. Chen?s optimal inequalities for statistical submanifolds involving Casorati curvature and the normalized scalar curavture in a statistical manifold of quasi constant curvature. The equality cases of these inequalities are also considered. Further, we provide some applications of our results. Moreover, as a new example we construct minimal statistical surface (statistical submanifold) of a statistical manifold of quasi constant curvature.

Recent Developments on the First Chen Inequality in Differential Geometry

“…In 2018, H. Aytimur and C. Özgür [198] studied statistical submanifolds of statistical manifolds of a quasi-constant curvature in which they obtained the Chen-Ricci inequality and generalized the Wingten inequality. In [199], P. Bansal, S. Uddin and M. H. Shahid derived a first Chen-type inequality for statistical submanifolds of a statistical manifold of a quasi-constant curvature. Also, G. He, J. Zhang and P. Zhao derived in [200] a first Chen-type inequality for a statistical submanifold of a statistical manifold M of a constant curvature, such that M admits a nonintegrable distribution on M with a constant rank.…”

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]).…”