MDPI - Publisher of Open Access Journals

22 pages, 341 KB

Open AccessArticle

Modular Invariance and Anomaly Cancellation Formulas for Fiber Bundles

by Jianyun Guan and Haiming Liu

Axioms 2025, 14(10), 740; https://doi.org/10.3390/axioms14100740 - 29 Sep 2025

Viewed by 159

By combining modular invariance of characteristic forms and the family index theory, we obtain some new anomaly cancellation formulas for any dimension under the not top degree component. For a fiber bundle of dimension

(4 l - 2)

, we obtain [...] Read more.

By combining modular invariance of characteristic forms and the family index theory, we obtain some new anomaly cancellation formulas for any dimension under the not top degree component. For a fiber bundle of dimension

(4 l - 2)

, we obtain the anomaly cancellation formulas for the determinant line bundle. For the fiber bundle with a dimension of

(4 l - 3)

, we derive the anomaly cancellation formulas of the index gerbes. For the fiber bundle of dimension

(4 l - 1)

, we obtain some results of the eta invariants. Moreover, we give some anomaly cancellation formulas of the Chen–Connes character and the higher elliptic genera. Full article

(This article belongs to the Section Geometry and Topology)

15 pages, 312 KB

Open AccessArticle

Inequality Constraints on Statistical Submanifolds of Norden-Golden-like Statistical Manifold

by Amit Kumar Rai, Majid Ali Choudhary, Mohammed Nisar and Foued Aloui

Symmetry 2025, 17(8), 1206; https://doi.org/10.3390/sym17081206 - 29 Jul 2025

Viewed by 396

Abstract

This paper explores novel inequalities for statistical submanifolds within the framework of the Norden golden-like statistical manifold. By leveraging the intrinsic properties of statistical manifolds and the structural richness of Norden golden geometry, we establish fundamental relationships between the intrinsic and extrinsic invariants [...] Read more.

This paper explores novel inequalities for statistical submanifolds within the framework of the Norden golden-like statistical manifold. By leveraging the intrinsic properties of statistical manifolds and the structural richness of Norden golden geometry, we establish fundamental relationships between the intrinsic and extrinsic invariants of submanifolds. The methodology involves deriving generalized Chen-type and

δ (2, 2)

curvature inequalities using curvature tensor analysis and dual affine connections. A concrete example is provided to verify the theoretical framework. The novelty of this work lies in extending classical curvature inequalities to a newly introduced statistical structure, thereby opening new perspectives in the study of geometric inequalities in information geometry and related mathematical physics contexts. Full article

(This article belongs to the Section Mathematics)

18 pages, 346 KB

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 905

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)

18 pages, 285 KB

Open AccessArticle

Chen-like Inequalities for Submanifolds in Kähler Manifolds Admitting Semi-Symmetric Non-Metric Connections

by Ion Mihai and Andreea Olteanu

Symmetry 2024, 16(10), 1401; https://doi.org/10.3390/sym16101401 - 21 Oct 2024

Viewed by 1347

Abstract

The geometry of submanifolds in Kähler manifolds is an important research topic. In the present paper, we study submanifolds in complex space forms admitting a semi-symmetric non-metric connection. We prove the Chen–Ricci inequality, Chen basic inequality, and a generalized Euler inequality for such [...] Read more.

The geometry of submanifolds in Kähler manifolds is an important research topic. In the present paper, we study submanifolds in complex space forms admitting a semi-symmetric non-metric connection. We prove the Chen–Ricci inequality, Chen basic inequality, and a generalized Euler inequality for such submanifolds. These inequalities provide estimations of the mean curvature (the main extrinsic invariants) in terms of intrinsic invariants: Ricci curvature, the Chen invariant, and scalar curvature. In the proofs, we use the sectional curvature of a semi-symmetric, non-metric connection recently defined by A. Mihai and the first author, as well as its properties. Full article

(This article belongs to the Special Issue Symmetry in Metric Spaces and Topology)

54 pages, 554 KB

Open AccessReview

A Comprehensive Review of Golden Riemannian Manifolds

by Bang-Yen Chen, Majid Ali Choudhary and Afshan Perween

Axioms 2024, 13(10), 724; https://doi.org/10.3390/axioms13100724 - 18 Oct 2024

Cited by 1 | Viewed by 5942

Abstract

In differential geometry, the concept of golden structure represents a compelling area with wide-ranging applications. The exploration of golden Riemannian manifolds was initiated by C. E. Hretcanu and M. Crasmareanu in 2008, following the principles of the golden structure. Subsequently, numerous researchers have [...] Read more.

In differential geometry, the concept of golden structure represents a compelling area with wide-ranging applications. The exploration of golden Riemannian manifolds was initiated by C. E. Hretcanu and M. Crasmareanu in 2008, following the principles of the golden structure. Subsequently, numerous researchers have contributed significant insights with respect to golden Riemannian manifolds. The purpose of this paper is to provide a comprehensive survey of research on golden Riemannian manifolds conducted over the past decade. Full article

(This article belongs to the Section Geometry and Topology)

16 pages, 299 KB

Open AccessArticle

Some Chen Inequalities for Submanifolds in Trans-Sasakian Manifolds Admitting a Semi-Symmetric Non-Metric Connection

by Mohammed Mohammed, Fortuné Massamba, Ion Mihai, Abd Elmotaleb A. M. A. Elamin and M. Saif Aldien

Axioms 2024, 13(3), 195; https://doi.org/10.3390/axioms13030195 - 15 Mar 2024

Cited by 1 | Viewed by 1958

Abstract

In the present article, we study submanifolds tangent to the Reeb vector field in trans-Sasakian manifolds. We prove Chen’s first inequality and the Chen–Ricci inequality, respectively, for such submanifolds in trans-Sasakian manifolds which admit a semi-symmetric non-metric connection. Moreover, a generalized Euler inequality [...] Read more.

In the present article, we study submanifolds tangent to the Reeb vector field in trans-Sasakian manifolds. We prove Chen’s first inequality and the Chen–Ricci inequality, respectively, for such submanifolds in trans-Sasakian manifolds which admit a semi-symmetric non-metric connection. Moreover, a generalized Euler inequality for special contact slant submanifolds in trans-Sasakian manifolds endowed with a semi-symmetric non-metric connection is obtained. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 2nd Edition)

20 pages, 345 KB

Open AccessArticle

An Invariant of Riemannian Type for Legendrian Warped Product Submanifolds of Sasakian Space Forms

by Fatemah Abdullah Alghamdi, Lamia Saeed Alqahtani, Ali H. Alkhaldi and Akram Ali

Mathematics 2023, 11(23), 4718; https://doi.org/10.3390/math11234718 - 21 Nov 2023

Cited by 2 | Viewed by 1142

Abstract

In the present paper, we investigate the geometry and topology of warped product Legendrian submanifolds in Sasakian space forms

D^{2 n + 1} (ϵ)

and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the [...] Read more.

In the present paper, we investigate the geometry and topology of warped product Legendrian submanifolds in Sasakian space forms

D^{2 n + 1} (ϵ)

and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the length of the warping functions. This inequality also involves intrinsic invariants (

δ

-invariant and sectional curvature). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the gradient Ricci curvature. Some new results on mean curvature vanishing are presented as a partial solution to the well-known problem given by S.S. Chern. Full article

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

50 pages, 653 KB

Open AccessReview

Recent Developments on the First Chen Inequality in Differential Geometry

by Bang-Yen Chen and Gabriel-Eduard Vîlcu

Mathematics 2023, 11(19), 4186; https://doi.org/10.3390/math11194186 - 6 Oct 2023

Cited by 3 | Viewed by 2117

Abstract

One of the most fundamental interests in submanifold theory is to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of submanifolds and find their applications. In this respect, the first author established, in 1993, a basic inequality involving [...] Read more.

One of the most fundamental interests in submanifold theory is to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of submanifolds and find their applications. In this respect, the first author established, in 1993, a basic inequality involving the first

δ

-invariant,

δ (2)

, and the squared mean curvature of submanifolds in real space forms, known today as the first Chen inequality or Chen’s first inequality. Since then, there have been many papers dealing with this inequality. The purpose of this article is, thus, to present a comprehensive survey on recent developments on this inequality performed by many geometers during the last three decades. Full article

(This article belongs to the Special Issue Differentiable Manifolds and Geometric Structures)

11 pages, 283 KB

Open AccessArticle

Geometric Inequalities for a Submanifold Equipped with Distributions

by Vladimir Rovenski

Mathematics 2022, 10(24), 4741; https://doi.org/10.3390/math10244741 - 14 Dec 2022

Cited by 3 | Viewed by 1423

Abstract

The article introduces invariants of a Riemannian manifold related to the mutual curvature of several pairwise orthogonal subspaces of a tangent bundle. In the case of one-dimensional subspaces, this curvature is equal to half the scalar curvature of the subspace spanned by them, [...] Read more.

The article introduces invariants of a Riemannian manifold related to the mutual curvature of several pairwise orthogonal subspaces of a tangent bundle. In the case of one-dimensional subspaces, this curvature is equal to half the scalar curvature of the subspace spanned by them, and in the case of complementary subspaces, this is the mixed scalar curvature. We compared our invariants with Chen invariants and proved geometric inequalities with intermediate mean curvature squared for a Riemannian submanifold. This gives sufficient conditions for the absence of minimal isometric immersions of Riemannian manifolds in a Euclidean space. As applications, geometric inequalities were obtained for isometric immersions of sub-Riemannian manifolds and Riemannian manifolds equipped with mutually orthogonal distributions. Full article

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

16 pages, 319 KB

Open AccessArticle

Improved Chen’s Inequalities for Submanifolds of Generalized Sasakian-Space-Forms

by Yanlin Li, Mohan Khatri, Jay Prakash Singh and Sudhakar K. Chaubey

Axioms 2022, 11(7), 324; https://doi.org/10.3390/axioms11070324 - 1 Jul 2022

Cited by 23 | Viewed by 2468

Abstract

In this article, we derive Chen’s inequalities involving Chen’s

δ

-invariant

δ_{M}

, Riemannian invariant

δ (m_{1}, \dots, m_{k})

, Ricci curvature, Riemannian invariant

Θ_{k} (2 \leq k \leq m)

, the scalar [...] Read more.

In this article, we derive Chen’s inequalities involving Chen’s

δ

-invariant

δ_{M}

, Riemannian invariant

δ (m_{1}, \dots, m_{k})

, Ricci curvature, Riemannian invariant

Θ_{k} (2 \leq k \leq m)

, the scalar curvature and the squared of the mean curvature for submanifolds of generalized Sasakian-space-forms endowed with a quarter-symmetric connection. As an application of the obtain inequality, we first derived the Chen inequality for the bi-slant submanifold of generalized Sasakian-space-forms. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application)

10 pages, 263 KB

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 1525

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)

38 pages, 518 KB

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 3369

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)

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 3298

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, 289 KB

Open AccessArticle

Relations between Extrinsic and Intrinsic Invariants of Statistical Submanifolds in Sasaki-Like Statistical Manifolds

by Hülya Aytimur, Adela Mihai and Cihan Özgür

Mathematics 2021, 9(11), 1285; https://doi.org/10.3390/math9111285 - 3 Jun 2021

Cited by 8 | Viewed by 2579

Abstract

The Chen first inequality and a Chen inequality for the

δ (2, 2)

-invariant on statistical submanifolds of Sasaki-like statistical manifolds, under a curvature condition, are obtained. Full article

10 pages, 247 KB

Open AccessArticle

A New Algebraic Inequality and Some Applications in Submanifold Theory

by Ion Mihai and Radu-Ioan Mihai

Mathematics 2021, 9(11), 1175; https://doi.org/10.3390/math9111175 - 23 May 2021

Cited by 3 | Viewed by 1818

Abstract

We give a simple proof of the Chen inequality involving the Chen invariant

δ (k)

of submanifolds in Riemannian space forms. We derive Chen’s first inequality and the Chen–Ricci inequality. Additionally, we establish a corresponding inequality for statistical submanifolds. Full article

(This article belongs to the Special Issue Applications of Inequalities and Functional Analysis)

Search Results (22)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (22)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI