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Mathematics
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Differential Geometric Structures and Their Applications
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Differential Geometric Structures and Their Applications

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

Deadline for manuscript submissions: 31 August 2025 | Viewed by 3703

Special Issue Editor

Prof. Dr. Vladimir Rovenski

E-Mail Website
Guest Editor
Department of Mathematics, University of Haifa, Mount Carmel, 3498838 Haifa, Israel
Interests: smooth manifold; submanifold; foliation; metric structure; curvature; tensor
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The Special Issue is devoted to Riemannian manifolds and submanifolds equipped with various additional geometric structures, such as foliation, distribution, almost product, almost contact, and complex structures. The topics covered in this Special Issue include the following: local and global differential geometry, curvature and topology of Riemannian manifolds and submanifolds (regular and with singularities); geometric inequalities related to extrinsic to intrinsic curvature invariants; variational problems for curvature functionals; almost contact (Sasakian, Kenmotsu, cosymplectic, etc.) structures and Ricci-type solitons; manifolds with density; etc.

Prof. Dr. Vladimir Rovenski
Guest Editor

Manuscript Submission Information

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Keywords

  • Riemannian manifold
  • submanifold
  • foliation
  • almost product manifold
  • almost contact manifold
  • second fundamental form
  • curvature
  • variation
  • Ricci-type soliton
  • Einstein-type metric
  • geometric analysis
  • geometric flow

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Published Papers (4 papers)

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Research

23 pages, 1537 KiB  
Article
CR-Selfdual Cubic Curves
by Mircea Crasmareanu, Cristina-Liliana Pripoae and Gabriel-Teodor Pripoae
Mathematics 2025, 13(2), 317; https://doi.org/10.3390/math13020317 - 19 Jan 2025
Cited by 1 | Viewed by 562
Abstract
We introduce a special class of cubic curves whose defining parameter satisfies a linear or quadratic equation provided by the values of a cross ratio. There are only seven such cubics and several properties of the real cubics in this class (some of [...] Read more.
We introduce a special class of cubic curves whose defining parameter satisfies a linear or quadratic equation provided by the values of a cross ratio. There are only seven such cubics and several properties of the real cubics in this class (some of them being elliptic curves) are discussed. Using the Möbius transformation, we extend this self-duality and obtain new families of remarkable complex cubics. In addition, we study (from the differential geometric viewpoint) the surface parameterized by all real cubic curves and we derive its curvature functions. As a by-product, we find a new classification of real Möbius transformations and some estimates for the number of vertices of real cubic curves. Full article
(This article belongs to the Special Issue Differential Geometric Structures and Their Applications)
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18 pages, 318 KiB  
Article
Spinor Equations of Smarandache Curves in E3
by Zeynep İsabeyoǧlu, Tülay Erişir and Ayşe Zeynep Azak
Mathematics 2024, 12(24), 4022; https://doi.org/10.3390/math12244022 - 22 Dec 2024
Viewed by 662
Abstract
This study examines the spinor representations of TN (tangent and normal), NB (normal and binormal), TB (tangent and binormal) and TNB (tangent, normal and binormal)–Smarandache curves in three-dimensional Euclidean space E3. Spinors are complex column vectors and move on Pauli spin [...] Read more.
This study examines the spinor representations of TN (tangent and normal), NB (normal and binormal), TB (tangent and binormal) and TNB (tangent, normal and binormal)–Smarandache curves in three-dimensional Euclidean space E3. Spinors are complex column vectors and move on Pauli spin matrices. Isotropic vectors in the C3 complex vector space form a two-dimensional surface in the C2 complex space. Additionally, each isotropic vector in C3 space corresponds to two vectors in C2 space, called spinors. Based on this information, our goal is to establish a relationship between curve theory in differential geometry and spinor space by matching a spinor with an isotropic vector and a real vector generated from the vectors of the Frenet–Serret frame of a curve in three-dimensional Euclidean space. Accordingly, we initially assume two spinors corresponding to the Frenet–Serret frames of the main curve and its (TN, NB, TB and TNB)–Smarandache curves. Then, we utilize the relationships between the Frenet frames of these curves to examine the connections between the two spinors corresponding to these curves. Thus, we give the relationships between spinors corresponding to these Smarandache curves. For this reason, this study creates a bridge between mathematics and physics. This study can also serve as a reference for new studies in geometry and physics as a geometric interpretation of a physical expression. Full article
(This article belongs to the Special Issue Differential Geometric Structures and Their Applications)
18 pages, 343 KiB  
Article
The Lichnerowicz-Type Laplacians: Vanishing Theorems for Their Kernels and Estimate Theorems for Their Smallest Eigenvalues
by Josef Mikeš, Sergey Stepanov and Irina Tsyganok
Mathematics 2024, 12(24), 3936; https://doi.org/10.3390/math12243936 - 14 Dec 2024
Viewed by 595
Abstract
In the present paper, we prove several vanishing theorems for the kernel of the Lichnerowicz-type Laplacian and provide estimates for its lowest eigenvalue on closed Riemannian manifolds. As an example of the Lichnerowicz-type Laplacian, we consider the Hodge–de Rham Laplacian acting on forms [...] Read more.
In the present paper, we prove several vanishing theorems for the kernel of the Lichnerowicz-type Laplacian and provide estimates for its lowest eigenvalue on closed Riemannian manifolds. As an example of the Lichnerowicz-type Laplacian, we consider the Hodge–de Rham Laplacian acting on forms and ordinary Lichnerowicz Laplacian acting on symmetric tensors. Additionally, we prove vanishing theorems for the null spaces of these Laplacians and find estimates for their lowest eigenvalues on closed Riemannian manifolds with suitably bounded curvature operators of the first kind, sectional and Ricci curvatures. Specifically, we will prove our version of the famous differential sphere theorem, which we will apply to the aforementioned problems concerning the ordinary Lichnerowicz Laplacian. Full article
(This article belongs to the Special Issue Differential Geometric Structures and Their Applications)
9 pages, 264 KiB  
Article
Weak Quasi-Contact Metric Manifolds and New Characteristics of K-Contact and Sasakian Manifolds
by Vladimir Rovenski
Mathematics 2024, 12(20), 3230; https://doi.org/10.3390/math12203230 - 15 Oct 2024
Viewed by 1361
Abstract
Quasi-contact metric manifolds (introduced by Y. Tashiro and then studied by several authors) are a natural extension of contact metric manifolds. Weak almost-contact metric manifolds, i.e., where the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, have [...] Read more.
Quasi-contact metric manifolds (introduced by Y. Tashiro and then studied by several authors) are a natural extension of contact metric manifolds. Weak almost-contact metric manifolds, i.e., where the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, have been defined by the author and R. Wolak. In this paper, we study a weak analogue of quasi-contact metric manifolds. Our main results generalize some well-known theorems and provide new criterions for K-contact and Sasakian manifolds in terms of conditions on the curvature tensor and other geometric objects associated with the weak quasi-contact metric structure. Full article
(This article belongs to the Special Issue Differential Geometric Structures and Their Applications)
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