Optimal inequalities for the Casorati curvatures of submanifolds of real space forms endowed with semi-symmetric metric connections

Abstract

In this paper, we prove two optimal inequalities involving the intrinsic scalar curvature and extrinsic Casorati curvature of submanifolds of real space forms endowed with a semi-symmetric metric connection. Moreover, we show that in both cases, the equality at all points characterizes the invariantly quasi-umbilical submanifolds. MSC: 53C40; 53B05

Inequalities for the Casorati Curvature of Statistical Manifolds in Holomorphic Statistical Manifolds of Constant Holomorphic Curvature

“…By virtue of Theorem 2, the generalized normalized δ-Casorati curvatures satisfy Inequalities (16) and (17). If the normalized δ-Casorati curvatures δ C (m − 1) and δ * C (m − 1), respectively,δ C (m − 1) and δ * C (m − 1) are involved, then we can state the following result.…”

“…In addition, we study the equality cases of Inequalities (16) and (17). First, we find out the critical points of P h c = (h 0 m+1 11 , h 0 m+1 12 , .…”

Casorati Inequalities for Submanifolds in a Riemannian Manifold of Quasi-Constant Curvature with a Semi-Symmetric Metric Connection

“…The proof of the inequalities in [4] is based on an optimization procedure by showing that the quadratic polynomial in the components of the second fundamental form is parabolic. And the above method was successfully applied to establish inequalities in terms of the Casorati curvatures for different submanifolds in various ambient spaces [5][6][7][8][9][10]. Recently, in [11][12][13], the authors obtained the corresponding Casorati inequalites by using Oprea's optimization methods on Riemannian submanifolds [14].…”

Inequalities for casorati curvatures of submanifolds in real space forms

“…The basic relationships discovered until now are inequalities. Recently the study of this topic has attracted a lot of attention [1,4,5,6,7,10,13,16,17,20].…”