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Open AccessEditorial

Differentiable Manifolds and Geometric Structures

by Adara M. Blaga

Mathematics 2025, 13(7), 1082; https://doi.org/10.3390/math13071082 - 26 Mar 2025

Cited by 3 | Viewed by 1026

This editorial presents 26 research articles published in the Special Issue entitled Differentiable Manifolds and Geometric Structures of the MDPI Mathematics journal, which covers a wide range of topics particularly from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the [...] Read more.

This editorial presents 26 research articles published in the Special Issue entitled Differentiable Manifolds and Geometric Structures of the MDPI Mathematics journal, which covers a wide range of topics particularly from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the latest achievements in different areas of differential geometry, among which is counted: the geometry of differentiable manifolds with curvature restrictions such as Golden space forms, Sasakian space forms; diffeological and affine connection spaces; Weingarten and Delaunay surfaces; Chen-type inequalities for submanifolds; statistical submersions; manifolds endowed with different geometric structures (Sasakian, weak nearly Sasakian, weak nearly cosymplectic, LP-Kenmotsu, paraquaternionic); solitons (almost Ricci solitons, almost Ricci–Bourguignon solitons, gradient r-almost Newton–Ricci–Yamabe solitons, statistical solitons, solitons with semi-symmetric connections); vector fields (projective, conformal, Killing, 2-Killing) [...] Full article

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

10 pages, 282 KB

Open AccessArticle

Weak Nearly Sasakian and Weak Nearly Cosymplectic Manifolds

by Vladimir Rovenski

Mathematics 2023, 11(20), 4377; https://doi.org/10.3390/math11204377 - 21 Oct 2023

Cited by 7 | Viewed by 2085

Abstract

Weak contact metric structures on a smooth manifold, introduced by V. Rovenski and R. Wolak in 2022, have provided new insight into the theory of classical structures. In this paper, we define new structures of this kind (called weak nearly Sasakian and weak [...] Read more.

Weak contact metric structures on a smooth manifold, introduced by V. Rovenski and R. Wolak in 2022, have provided new insight into the theory of classical structures. In this paper, we define new structures of this kind (called weak nearly Sasakian and weak nearly cosymplectic and nearly Kähler structures), study their geometry and give applications to Killing vector fields. We introduce weak nearly Kähler manifolds (generalizing nearly Kähler manifolds), characterize weak nearly Sasakian and weak nearly cosymplectic hypersurfaces in such Riemannian manifolds and prove that a weak nearly cosymplectic manifold with parallel Reeb vector field is locally the Riemannian product of a real line and a weak nearly Kähler manifold. Full article

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

26 pages, 341 KB

Open AccessArticle

On the Geometry of the Riemannian Curvature Tensor of Nearly Trans-Sasakian Manifolds

by Aligadzhi R. Rustanov

Axioms 2023, 12(9), 837; https://doi.org/10.3390/axioms12090837 - 29 Aug 2023

Cited by 1 | Viewed by 1703

Abstract

This paper presents the results of fundamental research into the geometry of the Riemannian curvature tensor of nearly trans-Sasakian manifolds. The components of the Riemannian curvature tensor on the space of the associated G-structure are counted, and the components of the Ricci tensor [...] Read more.

This paper presents the results of fundamental research into the geometry of the Riemannian curvature tensor of nearly trans-Sasakian manifolds. The components of the Riemannian curvature tensor on the space of the associated G-structure are counted, and the components of the Ricci tensor are calculated. Some identities are obtained that are satisfied by the Riemannian curvature tensors and the Ricci tensor. A number of properties are proved that characterize nearly trans-Sasakian manifolds with a closed contact form. The structure of nearly trans-Sasakian manifolds with a closed contact form is obtained. Several classes are singled out in terms of second-order differential-geometric invariants, and their local structure is obtained. The k-nullity distribution of a nearly trans-Sasakian manifold is studied. Full article

(This article belongs to the Special Issue Mathematical Analysis in Application to Solving Mathematical and Technical Problems)

14 pages, 282 KB

Open AccessArticle

Geometry of Harmonic Nearly Trans-Sasakian Manifolds

by Aligadzhi R. Rustanov

Axioms 2023, 12(8), 744; https://doi.org/10.3390/axioms12080744 - 28 Jul 2023

Cited by 3 | Viewed by 1523

Abstract

This paper considers a class of nearly trans-Sasakian manifolds. The local structure of nearly trans-Sasakian structures with a closed contact form and a closed Lee form is obtained. It is proved that the class of nearly trans-Sasakian manifolds with a closed contact form [...] Read more.

This paper considers a class of nearly trans-Sasakian manifolds. The local structure of nearly trans-Sasakian structures with a closed contact form and a closed Lee form is obtained. It is proved that the class of nearly trans-Sasakian manifolds with a closed contact form and a closed Lee form coincides with the class of almost contact metric manifolds with a closed contact form locally conformal to the closely cosymplectic manifolds. A wide class of harmonic nearly trans-Sasakian manifolds has been identified (i.e., nearly trans-Sasakian manifolds with a harmonic contact form) and an exhaustive description of the manifolds of this class is obtained. Also, examples of harmonic nearly trans-Sasakian manifolds are given. Full article

(This article belongs to the Special Issue Mathematical Analysis in Application to Solving Mathematical and Technical Problems)

11 pages, 288 KB

Open AccessArticle

Nearly Cosymplectic Manifolds of Constant Type

by Aligadzhi Rustanov

Axioms 2022, 11(4), 152; https://doi.org/10.3390/axioms11040152 - 25 Mar 2022

Cited by 5 | Viewed by 3200

Abstract

Fundamental identities characterizing a nearly cosymplectic structure and analytical expressions for the first and second structural tensors are obtained in this paper. An identity that is satisfied by the first structural tensor of a nearly cosymplectic structure is proved as well. A contact [...] Read more.

Fundamental identities characterizing a nearly cosymplectic structure and analytical expressions for the first and second structural tensors are obtained in this paper. An identity that is satisfied by the first structural tensor of a nearly cosymplectic structure is proved as well. A contact analog of nearly cosymplectic manifolds’ constancy of type is introduced in this paper. Pointwise constancy conditions of the type of nearly cosymplectic manifolds are obtained. It is proved that for nearly cosymplectic manifolds of dimension greater than three, pointwise constancy of type is equivalent to global constancy of type. A complete classification of nearly cosymplectic manifolds of constant type is obtained. It is also proved that a nearly cosymplectic manifold of dimension less than seven is a proper nearly cosymplectic manifold. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

11 pages, 279 KB

Open AccessArticle

A Geometric Obstruction for CR-Slant Warped Products in a Nearly Cosymplectic Manifold

by Siraj Uddin and M. Z. Ullah

Mathematics 2020, 8(9), 1622; https://doi.org/10.3390/math8091622 - 19 Sep 2020

Cited by 3 | Viewed by 2443

Abstract

In the early 20th century, B.-Y. Chen introduced the concept of CR-warped products and obtained several fundamental results, such as inequality for the length of second fundamental form. In this paper, we obtain B.-Y. Chen’s inequality for CR-slant warped products in nearly cosymplectic [...] Read more.

In the early 20th century, B.-Y. Chen introduced the concept of CR-warped products and obtained several fundamental results, such as inequality for the length of second fundamental form. In this paper, we obtain B.-Y. Chen’s inequality for CR-slant warped products in nearly cosymplectic manifolds, which are the more general classes of manifolds. The equality case of this inequality is also investigated. Furthermore, the inequality is discussed for some important subclasses of CR-slant warped products. Full article

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

11 pages, 772 KB

Open AccessArticle

On Generalized D-Conformal Deformations of Certain Almost Contact Metric Manifolds

by Nülifer Özdemir, Şirin Aktay and Mehmet Solgun

Mathematics 2019, 7(2), 168; https://doi.org/10.3390/math7020168 - 13 Feb 2019

Cited by 7 | Viewed by 3268

Abstract

In this work, we consider almost contact metric manifolds. We investigate the generalized D-conformal deformations of nearly K-cosymplectic, quasi-Sasakian and

β

-Kenmotsu manifolds. The new Levi–Civita covariant derivative of the new metric corresponding to deformed nearly K-cosymplectic, quasi-Sasakian and

β

-Kenmotsu manifolds are [...] Read more.

In this work, we consider almost contact metric manifolds. We investigate the generalized D-conformal deformations of nearly K-cosymplectic, quasi-Sasakian and

β

-Kenmotsu manifolds. The new Levi–Civita covariant derivative of the new metric corresponding to deformed nearly K-cosymplectic, quasi-Sasakian and

β

-Kenmotsu manifolds are obtained. Under some restrictions, deformed nearly K-cosymplectic, quasi-Sasakian and

β

-Kenmotsu manifolds are obtained. Then, the scalar curvature of these three classes of deformed manifolds are analyzed. Full article

10 pages, 282 KB

Open AccessArticle

The First Eigenvalue Estimates of Warped Product Pseudo-Slant Submanifolds

by Rifaqat Ali, Ali H. Alkhaldi, Akram Ali and Wan Ainun Mior Othman

Mathematics 2019, 7(2), 162; https://doi.org/10.3390/math7020162 - 11 Feb 2019

Cited by 1 | Viewed by 2479

Abstract

The aim of this paper is to construct a sharp general inequality for warped product pseudo-slant submanifold of the type

M = M_{⊥} \times_{f} M_{θ}

, in a nearly cosymplectic manifold, in terms of the warping function and the symmetric [...] Read more.

The aim of this paper is to construct a sharp general inequality for warped product pseudo-slant submanifold of the type

M = M_{⊥} \times_{f} M_{θ}

, in a nearly cosymplectic manifold, in terms of the warping function and the symmetric bilinear form h which is known as the second fundamental form. The equality cases are also discussed. As its application, we establish a bound for the first non-zero eigenvalue of the warping function whose base manifold is compact. Full article

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