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Mathematics
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Differential Geometry, Geometric Analysis and Their Related Applications
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Differential Geometry, Geometric Analysis and Their Related Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: 31 March 2025 | Viewed by 4087

Special Issue Editor

Prof. Dr. Mića S. Stanković

E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Sciences and Mathematics, University of Nis, Niš, Serbia
Interests: differential geometry; geodesic mappings; infinitesimal deformations of curves and surfaces in R3; tensor calculus; spaces with non symmetric affine connection; generalized Riemannian spaces; computer graphics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

As an important branch of mathematics, differential geometry provides a mathematical framework for understanding the geometric structures and properties of spaces. Differential geometry encompasses a wide range of topics and has applications in various areas of mathematics, physics, engineering, computer graphics, etc. This Special Issue is devoted to novel research on differential geometry, geometric analysis, and their wide-ranging applications. The topics covered in this Special Issue include, but are not limited to:

  1. Differential Geometry: Riemannian geometry, manifolds, Finsler geometry, symplectic geometry, contact geometry, complex and Kähler geometry, geodesic mappings, Minkowski spaces, almost geodesic mappings, etc.
  2. Geometric Analysis: minimal surfaces, geometric flows, geometric measure theory, geometric functional theory, geometric inequalities, such as the isoperimetric inequality, Sobolev inequalities, and the Minkowski inequality.
  3. Geometric Methods in Image Processing and Computer Vision: including topics such as geometric transformations, shape analysis, geometric modeling, and geometric methods for image registration and segmentation.
  4. Applications in Engineering and Applied Sciences.

All the related topics on the areas of differential geometry and geometric analysis will be of interest to this Special Issue.

Prof. Dr. Mića S. Stanković
Guest Editor

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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • differential geometry
  • geometric analysis
  • manifolds
  • Finsler geometry
  • symplectic geometry
  • contact geometry
  • complex and Kähler geometry
  • geodesic mappings
  • Minkowski spaces
  • geometric inequalities
  • geometric models
  • infinitesimal deformations of curves and surfaces
  • tensor calculus
  • spaces with non symmetric affine connection
  • generalized Riemannian spaces

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Published Papers (6 papers)

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10 pages, 241 KiB  
Article
Disaffinity Vectors on a Riemannian Manifold and Their Applications
by Sharief Deshmukh, Amira Ishan and Bang-Yen Chen
Mathematics 2024, 12(23), 3659; https://doi.org/10.3390/math12233659 - 22 Nov 2024
Viewed by 209
Abstract
A disaffinity vector on a Riemannian manifold (M,g) is a vector field whose affinity tensor vanishes. In this paper, we observe that nontrivial disaffinity functions offer obstruction to the topology of M and show that the existence of a [...] Read more.
A disaffinity vector on a Riemannian manifold (M,g) is a vector field whose affinity tensor vanishes. In this paper, we observe that nontrivial disaffinity functions offer obstruction to the topology of M and show that the existence of a nontrivial disaffinity function on M does not allow M to be compact. A characterization of the Euclidean space is also obtained by using nontrivial disaffinity functions. Further, we study properties of disaffinity vectors on M and prove that every Killing vector field is a disaffinity vector. Then, we prove that if the potential field ζ of a Ricci soliton M,g,ζ,λ is a disaffinity vector, then the scalar curvature is constant. As an application, we obtain conditions under which a Ricci soliton M,g,ζ,λ is trivial. Finally, we prove that a Yamabe soliton M,g,ξ,λ with a disaffinity potential field ξ is trivial. Full article
(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications)
11 pages, 258 KiB  
Article
Right Conoids Demonstrating a Time-like Axis within Minkowski Four-Dimensional Space
by Yanlin Li and Erhan Güler
Mathematics 2024, 12(15), 2421; https://doi.org/10.3390/math12152421 - 4 Aug 2024
Cited by 6 | Viewed by 675
Abstract
In the four-dimensional Minkowski space, hypersurfaces classified as right conoids with a time-like axis are introduced and studied. The computation of matrices associated with the fundamental form, the Gauss map, and the shape operator specific to these hypersurfaces is included in our analysis. [...] Read more.
In the four-dimensional Minkowski space, hypersurfaces classified as right conoids with a time-like axis are introduced and studied. The computation of matrices associated with the fundamental form, the Gauss map, and the shape operator specific to these hypersurfaces is included in our analysis. The intrinsic curvatures of these hypersurfaces are determined to provide a deeper understanding of their geometric properties. Additionally, the conditions required for these hypersurfaces to be minimal are established, and detailed calculations of the Laplace–Beltrami operator are performed. Illustrative examples are provided to enhance our comprehension of these concepts. Finally, the umbilical condition is examined to determine when these hypersurfaces become umbilic, and also the Willmore functional is explored. Full article
(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications)
16 pages, 284 KiB  
Article
Exploring Conformal Soliton Structures in Tangent Bundles with Ricci-Quarter Symmetric Metric Connections
by Yanlin Li, Aydin Gezer and Erkan Karakas
Mathematics 2024, 12(13), 2101; https://doi.org/10.3390/math12132101 - 4 Jul 2024
Cited by 8 | Viewed by 626
Abstract
In this study, we investigate the tangent bundle TM of an n-dimensional (pseudo-)Riemannian manifold M equipped with a Ricci-quarter symmetric metric connection ∇˜. Our primary goal is to establish the necessary and sufficient conditions for TM to exhibit [...] Read more.
In this study, we investigate the tangent bundle TM of an n-dimensional (pseudo-)Riemannian manifold M equipped with a Ricci-quarter symmetric metric connection ∇˜. Our primary goal is to establish the necessary and sufficient conditions for TM to exhibit characteristics of various solitons, specifically conformal Yamabe solitons, gradient conformal Yamabe solitons, conformal Ricci solitons, and gradient conformal Ricci solitons. We determine that for TM to be a conformal Yamabe soliton, the potential vector field must satisfy certain conditions when lifted vertically, horizontally, or completely from M to TM, alongside specific constraints on the conformal factor λ and the geometric properties of M. For gradient conformal Yamabe solitons, the conditions involve λ and the Hessian of the potential function. Similarly, for TM to be a conformal Ricci soliton, we identify conditions involving the lift of the potential vector field, the value of λ, and the curvature properties of M. For gradient conformal Ricci solitons, the criteria include the Hessian of the potential function and the Ricci curvature of M. These results enhance the understanding of the geometric properties of tangent bundles under Ricci-quarter symmetric metric connections and provide insights into their transition into various soliton states, contributing significantly to the field of differential geometry. Full article
(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications)
16 pages, 517 KiB  
Article
Alternative View of Inextensible Flows of Curves and Ruled Surfaces via Alternative Frame
by Ana Savić, Kemal Eren, Soley Ersoy and Vladimir Baltić
Mathematics 2024, 12(13), 2015; https://doi.org/10.3390/math12132015 - 28 Jun 2024
Viewed by 522
Abstract
In this paper, we present the evolutions of ruled surfaces generated by the principal normal, the principal normal’s derivative, and the Darboux vector fields along a space curve that are the elements of an alternative frame. The comprehension of an object’s rotational behavior [...] Read more.
In this paper, we present the evolutions of ruled surfaces generated by the principal normal, the principal normal’s derivative, and the Darboux vector fields along a space curve that are the elements of an alternative frame. The comprehension of an object’s rotational behavior is crucial knowledge relevant to various realms, and this can be accomplished by analyzing the Darboux vector along the path of a point on the object as it moves through space. In that regard, examining the evolutions of the ruled surfaces based on the changes in their directrices, including the Darboux vector in the alternative frame along a space curve, is significant. As the first step of this study, we express the evolution of the alternative frame elements of a space curve. Subsequently, the conditions for the ruled surfaces generated by them to be minimal, developable, and inextensible are investigated. These findings can allow some physical phenomena to be well understood through surface evolutions satisfying these conditions. In the final step, we provide graphical representations of some examples of inextensible ruled surfaces and curve evolutions. Full article
(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications)
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8 pages, 221 KiB  
Article
The Shape Operator of Real Hypersurfaces in S6(1)
by Djordje Kocić and Miroslava Antić
Mathematics 2024, 12(11), 1668; https://doi.org/10.3390/math12111668 - 27 May 2024
Viewed by 542
Abstract
The aim of the paper is to present two results concerning real hypersurfaces in the six-dimensional sphere S6(1). More precisely, we prove that real hypersurfaces with the Lie-parallel shape operator A must be totally geodesic hyperspheres. Additionally, we [...] Read more.
The aim of the paper is to present two results concerning real hypersurfaces in the six-dimensional sphere S6(1). More precisely, we prove that real hypersurfaces with the Lie-parallel shape operator A must be totally geodesic hyperspheres. Additionally, we classify real hypersurfaces in a nearly Kähler sphere S6(1) whose Lie derivative of the shape operator coincides with its covariant derivative. Full article
(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications)
12 pages, 268 KiB  
Article
On Curvature Pinching for Submanifolds with Parallel Normalized Mean Curvature Vector
by Juanru Gu and Yao Lu
Mathematics 2024, 12(11), 1633; https://doi.org/10.3390/math12111633 - 23 May 2024
Viewed by 713
Abstract
In this note, we investigate the pinching problem for oriented compact submanifolds of dimension n with parallel normalized mean curvature vector in a space form Fn+p(c). We first prove a codimension reduction theorem for submanifolds under [...] Read more.
In this note, we investigate the pinching problem for oriented compact submanifolds of dimension n with parallel normalized mean curvature vector in a space form Fn+p(c). We first prove a codimension reduction theorem for submanifolds under lower Ricci curvature bounds. Moreover, if the submanifolds have constant normalized scalar curvature R≥c, we obtain a classification theorem for submanifolds under lower Ricci curvature bounds. It should be emphasized that our Ricci pinching conditions are sharp for even n and p=2. Full article
(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications)
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