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Symmetry
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Symmetry and Its Application in Differential Geometry and Topology, 3rd Edition
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Symmetry and Its Application in Differential Geometry and Topology, 3rd Edition

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 December 2024) | Viewed by 11868

Special Issue Editors

Prof. Dr. Tiehong Zhao

E-Mail Website
Guest Editor
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
Interests: differential geometry
Special Issues, Collections and Topics in MDPI journals
Dr. Yanlin Li

E-Mail Website
Guest Editor
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
Interests: singularity theory; differential geometry
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Differential geometry is a branch of mathematics that has many applications not only in mathematics but in many other sciences, e.g., applications of the theory of curves and surfaces in the Euclidean plane and space. Geometry and Topology are quite related to Symmetry. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis. Differential geometry can be defined as the study of the geometry of differential manifolds, as well as of their submanifolds. In recent years, there has been a fast-growing interest in developing theories and tools for studying singular submanifolds. Because singular submanifolds are produced in physics, mechanics, and other application fields and are the breakthrough point to discover new problems. Therefore, it is of great scientific significance to study the geometric and topological properties of singular submanifolds. However, due to the existence of singular sets, the traditional analysis and geometric mathematical tools are no longer applicable, which makes the study of singular submanifolds difficult. In addition, applications of differential geometry and Topology can be found in almost any field of science, from biology to architecture. One of the most important applications of Topology is Topological Data Analysis (TDA). TDA combines ideas from Topology and also algebra, geometry, and analysis, with methods from statistics and computer science, for the purpose of analyzing contemporary data sets for which standard approaches are unsatisfactory. The motivating idea is that there is an underlying ''shape'' to the data and that new variants of some of the sophisticated tools of modern mathematics may be brought to bear to elucidate and learn from this structure. TDA has convincingly proved its utility in a wide range of applications in the life sciences, including in neuroscience, genomics, proteomics, evolution, and cancer biology, among other areas of research.

This Special Issue is intended to provide a series of papers focused on Symmetry and its applications of geometry and Topology, devoted to surveying the remarkable insights into many fields of sciences and exploring promising new developments.

Prof. Dr. Tiehong Zhao
Dr. Yanlin Li
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • singularity theory
  • morse theory/discrete Morse Theory
  • singularities
  • singular submanifolds
  • lightlike submanifolds
  • biharmonic submanifolds
  • warped product submanifolds
  • differentiable manifolds
  • Submanifold Theory
  • Legendrian duality
  • front and frontal
  • physics
  • statistics
  • topological data analysis
  • computational topology
  • applied topology and geometry
  • topological and geometric methods in data analysis
  • spectral and geometric methods in machine learning and data analysis
  • persistent homology and cohomology, and applications
  • neuroscience
  • cancer biology
  • genomics
  • other sciences

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Related Special Issue

Published Papers (10 papers)

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Research

11 pages, 263 KiB  
Article
Application of Differential Equations on the Ricci Curvature of Contact CR-Warped Product Submanifolds of S2n+1(1) with Semi-Symmetric Metric Connection
by Meraj Ali Khan, Amira A. Ishan, Ibrahim Al-Dayel and Khalid Masood
Symmetry 2024, 16(11), 1463; https://doi.org/10.3390/sym16111463 - 4 Nov 2024
Viewed by 795
Abstract
In this paper, we explore the uses of Obata’s differential equation in relation to the Ricci curvature of an odd-dimensional sphere that possesses a semi-symmetric metric connection. Specifically, we establish that, given certain conditions, the underlying submanifold can be identified as an isometric [...] Read more.
In this paper, we explore the uses of Obata’s differential equation in relation to the Ricci curvature of an odd-dimensional sphere that possesses a semi-symmetric metric connection. Specifically, we establish that, given certain conditions, the underlying submanifold can be identified as an isometric sphere. Additionally, we investigate the impact of specific differential equations on these submanifolds and demonstrate that, when certain geometric conditions are met, the base submanifold can be characterized as a special type of warped product. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 3rd Edition)
8 pages, 236 KiB  
Article
A Note on the Infinitesimal Bending of a Rectifying Curve
by Ştefan-Cezar Broscăţeanu, Adela Mihai and Andreea Olteanu
Symmetry 2024, 16(10), 1361; https://doi.org/10.3390/sym16101361 - 14 Oct 2024
Cited by 1 | Viewed by 721
Abstract
Both notions, of an infinitesimal bending of a curve and of a rectifying curve, play important roles in the theory of curves. In this short note, we begin the study of the infinitesimal bending of a rectifying curve. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 3rd Edition)
15 pages, 296 KiB  
Article
Some Aspects of Differential Topology of Subcartesian Spaces
by Liuyi Chen and Qianqian Xia
Symmetry 2024, 16(9), 1235; https://doi.org/10.3390/sym16091235 - 20 Sep 2024
Viewed by 1002
Abstract
In this paper, we investigate the differential topological properties of a large class of singular spaces: subcarteisan space. First, a minor further result on the partition of unity for differential spaces is derived. Second, the tubular neighborhood theorem for subcartesian spaces with constant [...] Read more.
In this paper, we investigate the differential topological properties of a large class of singular spaces: subcarteisan space. First, a minor further result on the partition of unity for differential spaces is derived. Second, the tubular neighborhood theorem for subcartesian spaces with constant structural dimensions is established. Third, the concept of Morse functions on smooth manifolds is generalized to differential spaces. For subcartesian space with constant structural dimension, a class of examples of Morse functions is provided. With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, it is proved that any smooth bounded function on this space can be approximated by Morse functions. The infinitesimal stability of Morse functions on subcartesian spaces is studied. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 3rd Edition)
15 pages, 265 KiB  
Article
Statistical Warped Product Immersions into Statistical Manifolds of (Quasi-)Constant Curvature
by Aliya Naaz Siddiqui, Meraj Ali Khan and Sudhakar Kumar Chaubey
Symmetry 2024, 16(6), 771; https://doi.org/10.3390/sym16060771 - 19 Jun 2024
Viewed by 1180
Abstract
Warped products provide an elegant and versatile framework for exploring and understanding a wide range of geometric structures. Their ability to combine two distinct manifolds through a warping function introduces a rich and diverse set of geometries, thus making them a powerful tool [...] Read more.
Warped products provide an elegant and versatile framework for exploring and understanding a wide range of geometric structures. Their ability to combine two distinct manifolds through a warping function introduces a rich and diverse set of geometries, thus making them a powerful tool in various mathematical, physical, and computational applications. This article addresses the central query related to warped product submanifolds in the context of statistics. It focuses on deriving two new and distinct inequalities for a statistical warped product submanifold in a statistical manifold of a constant (quasi-constant) curvature. This article then finishes with some concluding remarks. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 3rd Edition)
11 pages, 238 KiB  
Article
Semi-Symmetric Metric Connections and Homology of CR-Warped Product Submanifolds in a Complex Space Form Admitting a Concurrent Vector Field
by Meraj Ali Khan, Ibrahim Al-Dayel and Sudhakar Kumar Chaubey
Symmetry 2024, 16(6), 719; https://doi.org/10.3390/sym16060719 - 10 Jun 2024
Cited by 1 | Viewed by 1101
Abstract
In this paper, we conduct a thorough study of CR-warped product submanifolds in a Kaehler manifold, utilizing a semi-symmetric metric connection within the framework of warped product geometry. Our analysis yields fundamental and noteworthy results that illuminate the characteristics of these submanifolds. Additionally, [...] Read more.
In this paper, we conduct a thorough study of CR-warped product submanifolds in a Kaehler manifold, utilizing a semi-symmetric metric connection within the framework of warped product geometry. Our analysis yields fundamental and noteworthy results that illuminate the characteristics of these submanifolds. Additionally, we investigate the implications of our findings on the homology of these submanifolds, offering insights into their topological properties. Notably, we present a compelling proof demonstrating that, under a specific condition, stable currents cannot exist for these warped product submanifolds. Our research outcomes contribute significant knowledge concerning the stability and behavior of CR-warped product submanifolds equipped with a semi-symmetric metric connection. Furthermore, this work establishes a robust groundwork for future explorations and advancements in this particular field of study. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 3rd Edition)
15 pages, 326 KiB  
Article
Solitons of η-Ricci–Bourguignon Type on Submanifolds in (LCS)m Manifolds
by Lixu Yan, Vandana, Aliya Naaz Siddiqui, Halil Ibrahim Yoldas and Yanlin Li
Symmetry 2024, 16(6), 675; https://doi.org/10.3390/sym16060675 - 31 May 2024
Viewed by 884
Abstract
In this research article, we concentrate on the exploration of submanifolds in an (LCS)m-manifold B˜. We examine these submanifolds in the context of two distinct vector fields, namely, the characteristic vector field and the concurrent [...] Read more.
In this research article, we concentrate on the exploration of submanifolds in an (LCS)m-manifold B˜. We examine these submanifolds in the context of two distinct vector fields, namely, the characteristic vector field and the concurrent vector field. Initially, we consider some classifications of η-Ricci–Bourguignon (in short, η-RB) solitons on both invariant and anti-invariant submanifolds of B˜ employing the characteristic vector field. We establish several significant findings through this process. Furthermore, we investigate additional results by using η-RB solitons on invariant submanifolds of B˜ with concurrent vector fields, and discuss a supporting example. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 3rd Edition)
10 pages, 251 KiB  
Article
Ricci Solitons on Spacelike Hypersurfaces of Generalized Robertson–Walker Spacetimes
by Norah Alshehri and Mohammed Guediri
Symmetry 2024, 16(5), 601; https://doi.org/10.3390/sym16050601 - 13 May 2024
Cited by 1 | Viewed by 972
Abstract
In this paper, we investigate Ricci solitons on spacelike hypersurfaces in a special Lorentzian warped product manifold, the so-called generalized Robertson–Walker (GRW) spacetimes. Such spacetimes admit a natural form of symmetry which is represented by the conformal vector field f∂t, [...] Read more.
In this paper, we investigate Ricci solitons on spacelike hypersurfaces in a special Lorentzian warped product manifold, the so-called generalized Robertson–Walker (GRW) spacetimes. Such spacetimes admit a natural form of symmetry which is represented by the conformal vector field f∂t, where f is the warping function and ∂t is the unit timelike vector field tangent to the base (which is here a one-dimensional manifold). We use this symmetry to introduce some fundamental formulas related to the Ricci soliton structures and the Ricci curvature of the fiber, the warping function, and the shape operator of the immersion. We investigate different rigidity results for Ricci solitons on the slices, in addition to the totally umbilical spacelike supersurfaces of GRW. Furthermore, our study is focused on significant GRW spacetimes such as Einstein GRW spacetimes and those which obey the well-known null convergence condition (NCC). Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 3rd Edition)
16 pages, 290 KiB  
Article
Pinching Results for Doubly Warped Products’ Pointwise Bi-Slant Submanifolds in Locally Conformal Almost Cosymplectic Manifolds with a Quarter-Symmetric Connection
by Md Aquib, Ibrahim Al-Dayel, Mohd Aslam, Meraj Ali Khan and Mohammad Shuaib
Symmetry 2024, 16(5), 521; https://doi.org/10.3390/sym16050521 - 25 Apr 2024
Viewed by 976
Abstract
In this research paper, we establish geometric inequalities that characterize the relationship between the squared mean curvature and the warping functions of a doubly warped product pointwise bi-slant submanifold. Our investigation takes place in the context of locally conformal almost cosymplectic manifolds, which [...] Read more.
In this research paper, we establish geometric inequalities that characterize the relationship between the squared mean curvature and the warping functions of a doubly warped product pointwise bi-slant submanifold. Our investigation takes place in the context of locally conformal almost cosymplectic manifolds, which are equipped with a quarter-symmetric metric connection. We also consider the cases of equality in these inequalities. Additionally, we derive some geometric applications of our obtained results. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 3rd Edition)
16 pages, 318 KiB  
Article
Characterizations of Pointwise Hemi-Slant Warped Product Submanifolds in LCK Manifolds
by Fatimah Alghamdi
Symmetry 2024, 16(3), 281; https://doi.org/10.3390/sym16030281 - 29 Feb 2024
Cited by 1 | Viewed by 1226
Abstract
In this paper, we investigate the pointwise hemi-slant submanifolds of a locally conformal Kähler manifold and their warped products. Moreover, we derive the necessary and sufficient conditions for integrability and totally geodesic foliation. We establish characterization theorems for pointwise hemi-slant submanifolds. Several fundamental [...] Read more.
In this paper, we investigate the pointwise hemi-slant submanifolds of a locally conformal Kähler manifold and their warped products. Moreover, we derive the necessary and sufficient conditions for integrability and totally geodesic foliation. We establish characterization theorems for pointwise hemi-slant submanifolds. Several fundamental results that extend the CR submanifold warped product in Kähler manifolds are proven in this study. We also provide some non-trivial examples and applications. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 3rd Edition)
13 pages, 259 KiB  
Article
Contact CR-Warped Product Submanifold of a Sasakian Space Form with a Semi-Symmetric Metric Connection
by Meraj Ali Khan, Ibrahim Al-Dayel, Foued Aloui and Shyamal Kumar Hui
Symmetry 2024, 16(2), 190; https://doi.org/10.3390/sym16020190 - 6 Feb 2024
Cited by 1 | Viewed by 1238
Abstract
The main goal of this research paper is to investigate contact CR-warped product submanifolds within Sasakian space forms, utilizing a semi-symmetric metric connection. We conduct a comprehensive analysis of these submanifolds and establish several significant results. Additionally, we formulate an inequality that establishes [...] Read more.
The main goal of this research paper is to investigate contact CR-warped product submanifolds within Sasakian space forms, utilizing a semi-symmetric metric connection. We conduct a comprehensive analysis of these submanifolds and establish several significant results. Additionally, we formulate an inequality that establishes a relationship between the squared norm of the second fundamental form and the warping function. Lastly, we present a number of geometric applications derived from our findings. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 3rd Edition)
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