Optimal Inequalities for the Casorati Curvatures of Submanifolds of Generalized Space Forms Endowed With Semi-Symmetric Metric Connections

Abstract

Abstract. In this paper, we prove two optimal inequalities involving the intrinsic scalar curvature and extrinsic Casorati curvature of submanifolds of generalized space forms endowed with a semi-symmetric metric connection. Moreover, we also characterize those submanifolds for which the equality cases hold.

Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds

“…Several sharp inequalities between extrinsic and intrinsic curvatures for different submanifolds in real, complex, and quaternionic space forms endowed with various connections have been obtained (e.g., [14][15][16][17][18][19][20][21]). Such inequalities with a pair of conjugate affine connections involving the normalized scalar curvature of statistical submanifolds in different ambient spaces were obtained in [22][23][24][25][26].…”

Lower extremities for generalized normalized δ-Casorati curvatures of bi-slant submanifolds in generalized complex space forms

“…The geometrical meaning and the importance of the Casorati curvature has been discussed by distinguished geometers [6,12,19]. Therefore, it attracts the attention of geometers to obtain the optimal inequalities for the Casorati curvatures of the submanifolds of different ambient spaces [1,7,14].…”

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].…”

“…In [15,16], Mihai and Özgür established Chen inequalities for submanifolds of real, complex and Sasakian space forms endowed with semi-symmetric metric connections and in [17,18], Özgür and Murathan gave Chen inequalities for submanifolds of a locally conformal almost cosymplectic manifold and a cosymplectic space form endowed with semi-symmetric metric connections. On the other hand, Lee et al proved inequalities involving the Casorati curvature of submanifolds in real, complex and Sasakian space forms endowed with a semi-symmetric metric connection in [7,8]. In an earlier paper, Özgür established Chen inequalities for submanifolds in a Riemannian manifold of quasi-constant curvature [19].…”