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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)

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

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)

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)

11 pages, 237 KiB

Open AccessArticle

Pinching Theorems for a Vanishing C-Bochner Curvature Tensor

by Jae Won Lee and Chul Woo Lee

Mathematics 2018, 6(11), 231; https://doi.org/10.3390/math6110231 - 30 Oct 2018

Viewed by 2414

Abstract

The main purpose of this article is to construct inequalities between a main intrinsic invariant (the normalized scalar curvature) and an extrinsic invariant (the Casorati curvature) for some submanifolds in a Sasakian manifold with a zero C-Bochner tensor. Full article

(This article belongs to the Special Issue Differential Geometry)

18 pages, 274 KiB

Open AccessArticle

Pinching Theorems for Statistical Submanifolds in Sasaki-Like Statistical Space Forms

by Ali H. Alkhaldi, Mohd. Aquib, Aliya Naaz Siddiqui and Mohammad Hasan Shahid

Entropy 2018, 20(9), 690; https://doi.org/10.3390/e20090690 - 11 Sep 2018

Cited by 5 | Viewed by 3022

Abstract

In this paper, we obtain the upper bounds for the normalized

δ

-Casorati curvatures and generalized normalized

δ

-Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds with constant curvature. Further, we discuss the equality case of the inequalities. Moreover, we give the [...] Read more.

In this paper, we obtain the upper bounds for the normalized

δ

-Casorati curvatures and generalized normalized

δ

-Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds with constant curvature. Further, we discuss the equality case of the inequalities. Moreover, we give the necessary and sufficient condition for a Sasaki-like statistical manifold to be

η

-Einstein. Finally, we provide the condition under which the metric of Sasaki-like statistical manifolds with constant curvature is a solution of vacuum Einstein field equations. Full article

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