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

Abstract

In this article, we consider statistical submanifolds of Kenmotsu statistical manifolds of constant ϕ-sectional curvature. For such submanifold, we investigate curvature properties. We establish some inequalities involving the normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant). Moreover, we prove that the equality cases of the inequalities hold if and only if the imbedding curvature tensors h and h∗ of the submanifold (associated with the dual connections) satisfy h=−h∗, i.e., the submanifold is totally geodesic with respect to the Levi–Civita connection.

Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds

“…By subtracting (24) from (27), we can state the following result: Lemma 5. Let N = N 1 × f N 2 be an m-dimensional statistical warped product submanifold immersed into an n-dimensional statistical manifold of constant sectional curvaturec.…”

“…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].…”

Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications

“…which implies inequality. The equality sign holds in (2) if, and only if, the leaving terms in (23) and (24) imply that:…”

“…In [2,5,[20][21][22][23][24][25][26][27][28][29][30][31], the authors discuss the study of Einstein, contact metrics, and warped product manifolds for the above-mentioned problems. Furthermore, in regard to the collections of such inequalities, we referred to [12] and references therein.…”

Inequalities on Sasakian Statistical Manifolds in Terms of Casorati Curvatures

“…Moreover, Lee et al established that the normalized scalar curvature is bounded by Casorati curvatures of submanifolds in a statistical manifold of constant curvature [16]. In Kenmotsu statistical manifolds, Decu et al investigate curvature properties and establish optimizations in terms of a new extrinsic invariant (the normalized δ-Casorati curvature) and an intrinsic invariant (the scalar curvature) [17].…”