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18 pages, 301 KiB

Open AccessArticle

Casorati-Type Inequalities for Submanifolds in S-Space Forms with Semi-Symmetric Connection

by Md Aquib

Symmetry 2025, 17(7), 1100; https://doi.org/10.3390/sym17071100 - 9 Jul 2025

Viewed by 267

The primary aim of this paper is to establish two sharp geometric inequalities concerning submanifolds of S-space forms equipped with semi-symmetric metric connections (SSMCs). Specifically, we derive new inequalities involving the generalized normalized

δ

-Casorati curvatures [...] Read more.

The primary aim of this paper is to establish two sharp geometric inequalities concerning submanifolds of S-space forms equipped with semi-symmetric metric connections (SSMCs). Specifically, we derive new inequalities involving the generalized normalized

δ

-Casorati curvatures

δ_{c} (t; q_{1} + q_{2} - 1)

and

{\hat{δ}}_{c} (t; q_{1} + q_{2} - 1)

for bi-slant submanifolds. The cases in which equality holds are thoroughly examined, offering a deeper understanding of the geometric structure underlying such submanifolds. In addition, we present several immediate applications that highlight the relevance of our findings, and we support the article with illustrative examples. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)

15 pages, 268 KiB

Open AccessArticle

An Optimal Inequality for Warped Product Pointwise Semi-Slant Submanifolds in Complex Space Forms

by Md Aquib

Axioms 2025, 14(3), 213; https://doi.org/10.3390/axioms14030213 - 14 Mar 2025

Viewed by 385

Abstract

In this paper, we utilize advanced optimization techniques on Riemannian submanifolds to establish two distinct inequalities concerning the generalized normalized

δ

-Casorati curvatures of warped product pointwise semi-slant (WPPSS) submanifolds within complex space forms. We further identify the precise conditions under which these [...] Read more.

In this paper, we utilize advanced optimization techniques on Riemannian submanifolds to establish two distinct inequalities concerning the generalized normalized

δ

-Casorati curvatures of warped product pointwise semi-slant (WPPSS) submanifolds within complex space forms. We further identify the precise conditions under which these inequalities attain equality, providing valuable insights into their geometric and structural significance. Additionally, we also present results involving harmonic and Hessian functions, revealing a broader connection between curvature properties and analytic functions. Full article

(This article belongs to the Special Issue Advances in Geometry and Its Applications)

18 pages, 346 KiB

Open AccessFeature PaperArticle

Pinching Results for Submanifolds in Lorentzian–Sasakian Manifolds Endowed with a Semi-Symmetric Non-Metric Connection

by Mohammed Mohammed, Ion Mihai and Andreea Olteanu

Mathematics 2024, 12(23), 3651; https://doi.org/10.3390/math12233651 - 21 Nov 2024

Viewed by 844

Abstract

We establish an improved Chen inequality involving scalar curvature and mean curvature and geometric inequalities for Casorati curvatures, on slant submanifolds in a Lorentzian–Sasakian space form endowed with a semi-symmetric non-metric connection. Also, we present examples of slant submanifolds in a Lorentzian–Sasakian space [...] Read more.

We establish an improved Chen inequality involving scalar curvature and mean curvature and geometric inequalities for Casorati curvatures, on slant submanifolds in a Lorentzian–Sasakian space form endowed with a semi-symmetric non-metric connection. Also, we present examples of slant submanifolds in a Lorentzian–Sasakian space form. Full article

(This article belongs to the Special Issue Recent Studies in Differential Geometry and Its Applications)

20 pages, 363 KiB

Open AccessArticle

Optimal Inequalities on (α,β)-Type Almost Contact Manifold with the Schouten–Van Kampen Connection

by Mohd Danish Siddiqi and Ali H. Hakami

Axioms 2023, 12(12), 1082; https://doi.org/10.3390/axioms12121082 - 27 Nov 2023

Cited by 4 | Viewed by 1587

Abstract

In the current research, we develop optimal inequalities for submanifolds in trans-Sasakian manifolds or

(α, β)

-type almost contact manifolds endowed with the Schouten–Van Kampen connection (

S V K

-connection), including generalized normalized

δ

-Casorati Curvatures (

δ

-CC). [...] Read more.

In the current research, we develop optimal inequalities for submanifolds in trans-Sasakian manifolds or

(α, β)

-type almost contact manifolds endowed with the Schouten–Van Kampen connection (

S V K

-connection), including generalized normalized

δ

-Casorati Curvatures (

δ

-CC). We also discuss submanifolds on which the equality situations occur. Lastly, we provided an example derived from this research. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)

14 pages, 287 KiB

Open AccessArticle

Inequalities for the Generalized Normalized δ-Casorati Curvatures of Submanifolds in Golden Riemannian Manifolds

by Majid Ali Choudhary and Ion Mihai

Axioms 2023, 12(10), 952; https://doi.org/10.3390/axioms12100952 - 8 Oct 2023

Cited by 2 | Viewed by 1293

Abstract

In the present article, we consider submanifolds in golden Riemannian manifolds with constant golden sectional curvature. On such submanifolds, we prove geometric inequalities for the Casorati curvatures. The submanifolds meeting the equality cases are also described. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)

13 pages, 305 KiB

Open AccessArticle

Some Basic Inequalities on (ϵ)-Para Sasakian Manifold

by Majid Ali Choudhary, Mohammad Nazrul Islam Khan and Mohd Danish Siddiqi

Symmetry 2022, 14(12), 2585; https://doi.org/10.3390/sym14122585 - 7 Dec 2022

Cited by 6 | Viewed by 1756

Abstract

We propose fundamental inequalities for contact pseudo-slant submanifolds of

(ϵ)

-para Sasakian space form employing generalized normalized

δ

-Casorati curvature. We characterize submanifolds for which equality cases hold and illustrate the main result with some applications. Further, we have considered a [...] Read more.

We propose fundamental inequalities for contact pseudo-slant submanifolds of

(ϵ)

-para Sasakian space form employing generalized normalized

δ

-Casorati curvature. We characterize submanifolds for which equality cases hold and illustrate the main result with some applications. Further, we have considered a certain type of submanifold for a Ricci soliton and after computing its scalar curvature, developed an inequality to find correlations between intrinsic or extrinsic invariants. Full article

15 pages, 304 KiB

Open AccessArticle

Casorati Inequalities for Spacelike Submanifolds in Sasaki-like Statistical Manifolds with Semi-Symmetric Metric Connection

by Simona Decu

Mathematics 2022, 10(19), 3509; https://doi.org/10.3390/math10193509 - 26 Sep 2022

Viewed by 1244

Abstract

In this paper, we establish some inequalities between the normalized

δ

-Casorati curvatures and the scalar curvature (i.e., between extrinsic and intrinsic invariants) of spacelike statistical submanifolds in Sasaki-like statistical manifolds, endowed with a semi-symmetric metric connection. Moreover, we study the submanifolds satisfying [...] Read more.

In this paper, we establish some inequalities between the normalized

δ

-Casorati curvatures and the scalar curvature (i.e., between extrinsic and intrinsic invariants) of spacelike statistical submanifolds in Sasaki-like statistical manifolds, endowed with a semi-symmetric metric connection. Moreover, we study the submanifolds satisfying the equality cases of these inequalities. We also present an appropriate example. Full article

(This article belongs to the Special Issue Complex and Contact Manifolds II)

16 pages, 316 KiB

Open AccessFeature PaperArticle

Casorati Inequalities for Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature with Semi-Symmetric Metric Connection

by Simona Decu and Gabriel-Eduard Vîlcu

Entropy 2022, 24(6), 800; https://doi.org/10.3390/e24060800 - 8 Jun 2022

Cited by 5 | Viewed by 2000

Abstract

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely the normalized

δ

-Casorati curvatures and the scalar curvature of statistical submanifolds in Kenmotsu statistical manifolds of constant

ϕ

-sectional curvature that are endowed with semi-symmetric metric connection. Furthermore, [...] Read more.

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely the normalized

δ

-Casorati curvatures and the scalar curvature of statistical submanifolds in Kenmotsu statistical manifolds of constant

ϕ

-sectional curvature that are endowed with semi-symmetric metric connection. Furthermore, we investigate the equality cases of these inequalities. We also describe an illustrative example. Full article

10 pages, 263 KiB

Open AccessArticle

Generalized Wintgen Inequality for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature

by Aliya Naaz Siddiqui, Ali Hussain Alkhaldi and Lamia Saeed Alqahtani

Mathematics 2022, 10(10), 1727; https://doi.org/10.3390/math10101727 - 18 May 2022

Cited by 5 | Viewed by 1458

Abstract

The geometry of Hessian manifolds is a fruitful branch of physics, statistics, Kaehlerian and affine differential geometry. The study of inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature was truly initiated in 2018 by Mihai, A. and Mihai, I. who [...] Read more.

The geometry of Hessian manifolds is a fruitful branch of physics, statistics, Kaehlerian and affine differential geometry. The study of inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature was truly initiated in 2018 by Mihai, A. and Mihai, I. who dealt with Chen-Ricci and Euler inequalities. Later on, Siddiqui, A.N., Ahmad K. and Ozel C. came with the study of Casorati inequality for statistical submanifolds in the same ambient space by using algebraic technique. Also, Chen, B.-Y., Mihai, A. and Mihai, I. obtained a Chen first inequality for such submanifolds. In 2020, Mihai, A. and Mihai, I. studied the Chen inequality for

δ (2, 2)

-invariant. In the development of this topic, we establish the generalized Wintgen inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Some examples are also discussed at the end. Full article

(This article belongs to the Special Issue Geometry of Manifolds and Applications)

18 pages, 330 KiB

Open AccessArticle

Some Pinching Results for Bi-Slant Submanifolds in S-Space Forms

by Mohd Aquib, Meraj Ali Khan, Adela Mihai and Ion Mihai

Mathematics 2022, 10(9), 1538; https://doi.org/10.3390/math10091538 - 3 May 2022

Viewed by 1747

Abstract

The objective of the present article is to prove two geometric inequalities for submanifolds in S-space forms. First, we establish inequalities for the generalized normalized

δ

-Casorati curvatures for bi-slant submanifolds in S-space forms and then we derive the generalized Wintgen [...] Read more.

The objective of the present article is to prove two geometric inequalities for submanifolds in S-space forms. First, we establish inequalities for the generalized normalized

δ

-Casorati curvatures for bi-slant submanifolds in S-space forms and then we derive the generalized Wintgen inequality for Legendrian and bi-slant submanifolds in the same ambient space. We also discuss the equality cases of the inequalities. Further, we provide some immediate geometric applications of the results. Finally, we construct some examples of slant and Legendrian submanifolds, respectively. Full article

(This article belongs to the Special Issue Complex and Contact Manifolds II)

38 pages, 518 KiB

Open AccessFeature PaperReview

Differential Geometry of Submanifolds in Complex Space Forms Involving δ-Invariants

by Bang-Yen Chen, Adara M. Blaga and Gabriel-Eduard Vîlcu

Mathematics 2022, 10(4), 591; https://doi.org/10.3390/math10040591 - 14 Feb 2022

Cited by 10 | Viewed by 3188

Abstract

One of the fundamental problems in the theory of submanifolds is to establish optimal relationships between intrinsic and extrinsic invariants for submanifolds. In order to establish such relations, the first author introduced in the 1990s the notion of

δ

-invariants for Riemannian manifolds, [...] Read more.

One of the fundamental problems in the theory of submanifolds is to establish optimal relationships between intrinsic and extrinsic invariants for submanifolds. In order to establish such relations, the first author introduced in the 1990s the notion of

δ

-invariants for Riemannian manifolds, which are different in nature from the classical curvature invariants. The earlier results on

δ

-invariants and their applications have been summarized in the first author’s book published in 2011 Pseudo-Riemannian Geometry, δ-Invariants and Applications (ISBN: 978-981-4329-63-7). In this survey, we present a comprehensive account of the development of the differential geometry of submanifolds in complex space forms involving the

δ

-invariants done mostly after the publication of the book. Full article

(This article belongs to the Special Issue Geometry of Manifolds and Applications)

15 pages, 296 KiB

Open AccessArticle

Bounds for Statistical Curvatures of Submanifolds in Kenmotsu-like Statistical Manifolds

by Aliya Naaz Siddiqui, Mohd Danish Siddiqi and Ali Hussain Alkhaldi

Mathematics 2022, 10(2), 176; https://doi.org/10.3390/math10020176 - 6 Jan 2022

Cited by 4 | Viewed by 1444

Abstract

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized [...] Read more.

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized normalized Casorati curvatures. We also define the second fundamental form of those submanifolds that satisfy the equality condition. On Legendrian submanifolds of Kenmotsu-like statistical manifolds, we discuss a conjecture for Wintgen inequality. At the end, some immediate geometric consequences are stated. Full article

(This article belongs to the Special Issue Geometry of Manifolds and Applications)

13 pages, 296 KiB

Open AccessArticle

Inequalities for the Casorati Curvature of Totally Real Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms

by Bang-Yen Chen, Simona Decu and Gabriel-Eduard Vîlcu

Entropy 2021, 23(11), 1399; https://doi.org/10.3390/e23111399 - 25 Oct 2021

Cited by 10 | Viewed by 1897

Abstract

The purpose of this article is to establish some inequalities concerning the normalized

δ

-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of totally real spacelike submanifolds in statistical manifolds of the type para-Kähler space form. Moreover, this study is focused [...] Read more.

The purpose of this article is to establish some inequalities concerning the normalized

δ

-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of totally real spacelike submanifolds in statistical manifolds of the type para-Kähler space form. Moreover, this study is focused on the equality cases in these inequalities. Some examples are also provided. Full article

18 pages, 303 KiB

Open AccessArticle

On Golden Lorentzian Manifolds Equipped with Generalized Symmetric Metric Connection

by Majid Ali Choudhary, Khaled Mohamed Khedher, Oğuzhan Bahadır and Mohd Danish Siddiqi

Mathematics 2021, 9(19), 2430; https://doi.org/10.3390/math9192430 - 30 Sep 2021

Cited by 9 | Viewed by 1975

Abstract

This research deals with the generalized symmetric metric U-connection defined on golden Lorentzian manifolds. We also derive sharp geometric inequalities that involve generalized normalized

δ

-Casorati curvatures for submanifolds of golden Lorentzian manifolds equipped with generalized symmetric metric U-connection. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

13 pages, 267 KiB

Open AccessArticle

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

by Simona Decu, Stefan Haesen and Leopold Verstraelen

Mathematics 2020, 8(2), 251; https://doi.org/10.3390/math8020251 - 14 Feb 2020

Cited by 14 | Viewed by 2618

Abstract

In this paper, we prove some inequalities in terms of the normalized

δ

-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of statistical submanifolds in holomorphic statistical manifolds with constant holomorphic sectional curvature. Moreover, we study the equality cases of such [...] Read more.

In this paper, we prove some inequalities in terms of the normalized

δ

-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of statistical submanifolds in holomorphic statistical manifolds with constant holomorphic sectional curvature. Moreover, we study the equality cases of such inequalities. An example on these submanifolds is presented. Full article

(This article belongs to the Special Issue Inequalities in Geometry and Applications)

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