Optimal inequalities for the normalized δ-Casorati curvatures of submanifolds in Kenmotsu space forms

Abstract

In this paper, we establish two sharp inequalities for the normalized

Optimal casorati inequalities on bi-slant submanifolds of generalized sasakian space forms

“…By the definition of δ C (n), we can obtain our desired inequality (11). From (20) and 21, we conclude that the equality sign holds in the inequality (11) if and only if the submanifold M is invariantly quasi-umbilical with trivial normal connection in M , such that with respect to suitable orthonormal tangent and normal orthonormal frames, the shape operators take the form of (13).…”

“…Y. Chen [5] introduced the new types of Riemannian invariants, known in the literature as Chen invariants and obtained general optimal inequalities consisting of the new intrinsic invariants and the main extrinsic invariants for any Riemannian submanifolds. It was the starting point of the theory of Chen invariants, which are one of the most interesting research topics in differential geometry [11,12,13,14,16,17,18,19,24]. Instead of concentrating on the sectional curvature with the extrinsic squared mean curvature, the Casorati curvature of a submanifold in a Riemannian manifold was considered as an extrinsic invariant defined as the normalized square of the length of the second fundamental form.…”

Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature

“…Suppose that the hyperplane L is spanned by the tangent vectors e 1 , ..., e m , avoiding loss of generality. Then, from Equations (20) and 21, we derive that P has the expression…”

“…Submanifolds for which these equalities hold are called Casorati ideal submanifolds. Recently, Lee et al [20] studied optimal inequalities in terms of δ-Casorati curvatures of submanifolds in Kenmotsu space forms. We recall that the Kenmotsu geometry is an area of contact geometry initiated by Kenmotsu in [21], with many applications, e.g., in physics (geometrical optics, classical mechanics, thermodynamics, geometric quantization) and control theory [22].…”

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].…”