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15 pages, 328 KB

Open AccessArticle

Gradient Expanding Ricci Solitons Type Inequalities on Submanifolds in Quaternion Kaehler Manifolds with Bi-Slant Factor

by Ali H. Hakami and Mohd Danish Siddiqi

Mathematics 2026, 14(2), 357; https://doi.org/10.3390/math14020357 - 21 Jan 2026

Viewed by 106

Abstract

In this article, we study the Ricci soliton on quaternion bi-slant submanifolds of quaternion Kaehler manifolds. We obtain a lower-bound-type inequality in terms of expanding gradient Ricci solitons with a gradient-type vector field for the quaternion bi-slant submanifold of quaternion Kaehler manifolds. Additionally, [...] Read more.

In this article, we study the Ricci soliton on quaternion bi-slant submanifolds of quaternion Kaehler manifolds. We obtain a lower-bound-type inequality in terms of expanding gradient Ricci solitons with a gradient-type vector field for the quaternion bi-slant submanifold of quaternion Kaehler manifolds. Additionally, we derive a series of lower-bound-type inequalities for semi-slant submanifolds,

C R

-submanifolds, hemi-slant submanifolds, slant submanifolds, invariant anti-invaraint submanifolds and totally real submanifolds in the same quaternion Kaehler manifolds. Finally, we discuss a double inequality for expanding gradient Ricci solitons on submanifolds in quaternion Kaehler manifolds and extend the same double inequality in terms of gradient Ricci solitons with a scalar concircular field on semi-slant, quaternion

C R

-submanifolds of quaternion Kaehler manifolds. Full article

(This article belongs to the Special Issue Submanifolds in Metric Manifolds, 2nd Edition)

30 pages, 435 KB

Open AccessArticle

Classification of Four-Dimensional CR Submanifolds of the Homogenous Nearly Kähler

S^{3} \times S^{3}

Which Almost Complex Distribution Is Almost Product Orthogonal on Itself

by Nataša Djurdjević

Mathematics 2025, 13(16), 2638; https://doi.org/10.3390/math13162638 - 17 Aug 2025

Viewed by 659

Abstract

The product manifold

S^{3} \times S^{3}

, which belongs to the homogenous six-dimensional nearly Kähler manifolds, admits two structures, the almost complex structure J and the almost product structure P. The investigation of embeddings of different classes of CR submanifolds [...] Read more.

The product manifold

S^{3} \times S^{3}

, which belongs to the homogenous six-dimensional nearly Kähler manifolds, admits two structures, the almost complex structure J and the almost product structure P. The investigation of embeddings of different classes of CR submanifolds of

S^{3} \times S^{3}

was started some time ago by investigating three-dimensional CR submanifolds. It resulted that the almost product structure P is very important for the study of CR submanifolds of

S^{3} \times S^{3}

, since submanifolds characterized by different actions of the almost product structure on base vector fields often appear as a result of the study of some specific types of CR submanifolds. Therefore, the investigation of four-dimensional CR submanifolds of

S^{3} \times S^{3}

is initiated in this article. The main result is the classification of four-dimensional CR submanifolds of

S^{3} \times S^{3}

, whose almost complex distribution

D_{1}

is almost product orthogonal on itself. First, it was proved that such submanifolds have a non-integrable almost complex distribution, and then it was proved that these submanifolds are locally product manifolds of curves and three-dimensional CR submanifolds of

S^{3} \times S^{3}

of the same type, and they were therefore constructed in this way. Full article

(This article belongs to the Special Issue Submanifolds in Metric Manifolds, 2nd Edition)

12 pages, 253 KB

Open AccessArticle

Pinching Results on Totally Real Submanifolds of a Locally Conformal Kähler Manifolds

by Noura M. Alhouiti, Ali H. Alkhaldi, Akram Ali and Piscoran Laurian-Ioan

Mathematics 2025, 13(10), 1682; https://doi.org/10.3390/math13101682 - 21 May 2025

Viewed by 603

Abstract

This paper investigates the relationship between pseudo-umbilical and minimal totally real submanifolds in locally conformal Kähler space forms. Some rigidity theorems and an integral inequality are obtained using the moving-frame method and the DDVV inequality. Our results extend this line of previous research. [...] Read more.

This paper investigates the relationship between pseudo-umbilical and minimal totally real submanifolds in locally conformal Kähler space forms. Some rigidity theorems and an integral inequality are obtained using the moving-frame method and the DDVV inequality. Our results extend this line of previous research. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

11 pages, 244 KB

Open AccessArticle

Optimal Inequalities Characterizing Totally Real Submanifolds in Quaternionic Space Form

by Fatimah Alghamdi and Akram Ali

Mathematics 2025, 13(10), 1643; https://doi.org/10.3390/math13101643 - 17 May 2025

Viewed by 563

Abstract

In the present paper, we investigate some pinching inequalities on the scalar curvature of a totally real submanifold in quaternionic space form that leads to a topological conclusion of the submanifold. In addition, we construct another inequality which includes the mean curvature and [...] Read more.

In the present paper, we investigate some pinching inequalities on the scalar curvature of a totally real submanifold in quaternionic space form that leads to a topological conclusion of the submanifold. In addition, we construct another inequality which includes the mean curvature and the length of the second fundamental form. Full article

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

11 pages, 243 KB

Open AccessArticle

Eigenvalues for Laplacian Operator on Submanifolds in Locally Conformal Kaehler Space Forms

by Noura M. Alhouiti, Ali H. Alkhaldi, Akram Ali, Fatemah Mofarreh and Piscoran Laurian-Ioan

Axioms 2025, 14(5), 356; https://doi.org/10.3390/axioms14050356 - 8 May 2025

Cited by 1 | Viewed by 890

Abstract

This paper investigates totally real submanifolds in a locally conformal Kaehler space form. Using the moving-frame method and constant mean curvature, we obtain the upper and lower bounds of the first eigenvalue for totally real submanifolds in a locally conformal Kaehler space form. [...] Read more.

This paper investigates totally real submanifolds in a locally conformal Kaehler space form. Using the moving-frame method and constant mean curvature, we obtain the upper and lower bounds of the first eigenvalue for totally real submanifolds in a locally conformal Kaehler space form. We discussed the integral inequalities and their properties. Some previous results are generalized from our results. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

18 pages, 274 KB

Open AccessArticle

Quaternion Statistical Submanifolds and Submersions

by Aliya Naaz Siddiqui and Fatimah Alghamdi

Mathematics 2025, 13(1), 53; https://doi.org/10.3390/math13010053 - 27 Dec 2024

Viewed by 912

Abstract

This paper aims to develop a general theory of quaternion Kahlerian statistical manifolds and to study quaternion CR-statistical submanifolds in such ambient manifolds. It extends the existing theories of quaternion submanifolds and totally real submanifolds. Additionally, the work examines quaternion Kahlerian statistical submersions, [...] Read more.

This paper aims to develop a general theory of quaternion Kahlerian statistical manifolds and to study quaternion CR-statistical submanifolds in such ambient manifolds. It extends the existing theories of quaternion submanifolds and totally real submanifolds. Additionally, the work examines quaternion Kahlerian statistical submersions, including illustrative examples. The exploration also includes an analysis of the total space and fibers under certain conditions with example(s) in support. Moreover, Chen–Ricci inequality on the vertical distribution is derived for quaternion Kahlerian statistical submersions from quaternion Kahlerian statistical manifolds. Full article

(This article belongs to the Section B: Geometry and Topology)

15 pages, 275 KB

Open AccessArticle

DDVV Inequality on Submanifolds Coupled with a Slant Factor in Quaternionic Kaehler Manifolds

by Rawan Bossly, Majid Ali Choudhary, Mohd Danish Siddiqi, Oḡuzhan Bahadır and Mehmet Gülbahar

Axioms 2025, 14(1), 6; https://doi.org/10.3390/axioms14010006 - 26 Dec 2024

Viewed by 1439

Abstract

This work aims to provide generalized Wintgen inequalities for slant submanifolds embedded in quaternionic space forms, taking into consideration both semi-symmetric metric and semi-symmetric non-metric connections. Moreover, we discuss the same inequality for totally real, anti-invariant, and invariant submanifolds on quaternionic space forms, [...] Read more.

This work aims to provide generalized Wintgen inequalities for slant submanifolds embedded in quaternionic space forms, taking into consideration both semi-symmetric metric and semi-symmetric non-metric connections. Moreover, we discuss the same inequality for totally real, anti-invariant, and invariant submanifolds on quaternionic space forms, endowed with both semi-symmetric metric and semi-symmetric non-metric connections. We also characterized the equality case through specific forms of shape operators of Wintgen inequalities for these classes of submanifolds in quaternionic space forms, admitting a semi-symmetric metric connection and a semi-symmetric non-metric connection. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

11 pages, 306 KB

Open AccessArticle

Geometry of CR-Slant Warped Products in Nearly Kaehler Manifolds

by Siraj Uddin, Bang-Yen Chen and Rawan Bossly

Mathematics 2023, 11(12), 2600; https://doi.org/10.3390/math11122600 - 7 Jun 2023

Cited by 3 | Viewed by 1704

Abstract

Recently, we studied CR-slant warped products

B_{1} \times_{f} M_{⊥}

, where

B_{1} = M_{T} \times M_{θ}

is the Riemannian product of holomorphic and proper slant submanifolds and

M_{⊥}

is a totally real submanifold in a nearly [...] Read more.

Recently, we studied CR-slant warped products

B_{1} \times_{f} M_{⊥}

, where

B_{1} = M_{T} \times M_{θ}

is the Riemannian product of holomorphic and proper slant submanifolds and

M_{⊥}

is a totally real submanifold in a nearly Kaehler manifold. In the continuation, in this paper, we study

B_{2} \times_{f} M_{θ}

, where

B_{2} = M_{T} \times M_{⊥}

is a CR-product of a nearly Kaehler manifold and establish Chen’s inequality for the squared norm of the second fundamental form. Some special cases of Chen’s inequality are given. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Submanifolds)

12 pages, 274 KB

Open AccessEditor’s ChoiceArticle

Chen Inequalities for Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms

by Simona Decu and Stefan Haesen

Mathematics 2022, 10(3), 330; https://doi.org/10.3390/math10030330 - 21 Jan 2022

Cited by 9 | Viewed by 3438

Abstract

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely involving the Chen first invariant and the mean curvature of totally real and holomorphic spacelike submanifolds in statistical manifolds of type para-Kähler space forms. Furthermore, we investigate the equality [...] Read more.

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely involving the Chen first invariant and the mean curvature of totally real and holomorphic spacelike submanifolds in statistical manifolds of type para-Kähler space forms. Furthermore, we investigate the equality cases of these inequalities. As illustrations of the applications of the above inequalities, we consider a few examples. Full article

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

13 pages, 296 KB

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 12 | Viewed by 2098

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

11 pages, 259 KB

Open AccessArticle

Submanifolds in Normal Complex Contact Manifolds

by Adela Mihai and Ion Mihai

Mathematics 2019, 7(12), 1195; https://doi.org/10.3390/math7121195 - 5 Dec 2019

Viewed by 2531

Abstract

In the present article we initiate the study of submanifolds in normal complex contact metric manifolds. We define invariant and anti-invariant (

C C

-totally real) submanifolds in such manifolds and start the study of their basic properties. Also, we establish the Chen [...] Read more.

In the present article we initiate the study of submanifolds in normal complex contact metric manifolds. We define invariant and anti-invariant (

C C

-totally real) submanifolds in such manifolds and start the study of their basic properties. Also, we establish the Chen first inequality and Chen inequality for the invariant

δ (2, 2)

for

C C

-totally real submanifolds in a normal complex contact space form and characterize the equality cases. We also prove the minimality of

C C

-totally real submanifolds of maximum dimension satisfying the equalities. Full article

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

64 pages, 661 KB

Open AccessReview

A Comprehensive Survey on Parallel Submanifolds in Riemannian and Pseudo-Riemannian Manifolds

by Bang-Yen Chen

Axioms 2019, 8(4), 120; https://doi.org/10.3390/axioms8040120 - 30 Oct 2019

Cited by 2 | Viewed by 4527

Abstract

A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden–Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. [...] Read more.

A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden–Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifolds form an important class of Riemannian submanifolds since extrinsic invariants of a parallel submanifold do not vary from point to point. In this paper, we provide a comprehensive survey on this important class of submanifolds. Full article

(This article belongs to the Special Issue Geometric Analysis and Mathematical Physics)

15 pages, 284 KB

Open AccessFeature PaperArticle

Bi-Slant Submanifolds of Para Hermitian Manifolds

by Pablo Alegre and Alfonso Carriazo

Mathematics 2019, 7(7), 618; https://doi.org/10.3390/math7070618 - 11 Jul 2019

Cited by 17 | Viewed by 3229 | Correction

Abstract

In this paper, we introduce the notion of bi-slant submanifolds of a para Hermitian manifold. They naturally englobe CR, semi-slant, and hemi-slant submanifolds. We study their first properties and present a whole gallery of examples. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

10 pages, 248 KB

Open AccessArticle

Inequalities on Sasakian Statistical Manifolds in Terms of Casorati Curvatures

by Chul Woo Lee and Jae Won Lee

Mathematics 2018, 6(11), 259; https://doi.org/10.3390/math6110259 - 17 Nov 2018

Cited by 5 | Viewed by 2850

Abstract

A statistical structure is considered as a generalization of a pair of a Riemannian metric and its Levi-Civita connection. With a pair of conjugate connections ∇ and

\nabla^{*}

in the Sasakian statistical structure, we provide the normalized scalar curvature which is bounded [...] Read more.

A statistical structure is considered as a generalization of a pair of a Riemannian metric and its Levi-Civita connection. With a pair of conjugate connections ∇ and

\nabla^{*}

in the Sasakian statistical structure, we provide the normalized scalar curvature which is bounded above from Casorati curvatures on C-totally real (Legendrian and slant) submanifolds of a Sasakian statistical manifold of constant

φ

-sectional curvature. In addition, we give examples to show that the total space is a sphere. Full article

(This article belongs to the Special Issue Differential Geometry)

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