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6 pages, 177 KiB

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

Viewed by 425

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 KiB

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 6 | Viewed by 1626

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 KiB

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

Viewed by 1273

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 KiB

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 2 | Viewed by 1176

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)

19 pages, 304 KiB

Open AccessArticle

On Nearly Sasakian and Nearly Kähler Statistical Manifolds

by Siraj Uddin, Esmaeil Peyghan, Leila Nourmohammadifar and Rawan Bossly

Mathematics 2023, 11(12), 2644; https://doi.org/10.3390/math11122644 - 9 Jun 2023

Cited by 5 | Viewed by 1515

Abstract

In this paper, we introduce the notions of nearly Sasakian and nearly Kähler statistical structures with a non-trivial example. The conditions for a real hypersurface in a nearly Kähler statistical manifold to admit a nearly Sasakian statistical structure are given. We also study [...] Read more.

In this paper, we introduce the notions of nearly Sasakian and nearly Kähler statistical structures with a non-trivial example. The conditions for a real hypersurface in a nearly Kähler statistical manifold to admit a nearly Sasakian statistical structure are given. We also study invariant and anti-invariant statistical submanifolds of nearly Sasakian statistical manifolds. Finally, some conditions under which such a submanifold of a nearly Sasakian statistical manifold is itself a nearly Sasakian statistical manifold are given. Full article

(This article belongs to the Special Issue Submanifolds in Metric Manifolds)

20 pages, 330 KiB

Open AccessArticle

A Note on Nearly Sasakian Manifolds

by Fortuné Massamba and Arthur Nzunogera

Mathematics 2023, 11(12), 2634; https://doi.org/10.3390/math11122634 - 9 Jun 2023

Cited by 3 | Viewed by 1584

Abstract

A class of nearly Sasakian manifolds is considered in this paper. We discuss the geometric effects of some symmetries on such manifolds and show, under a certain condition, that the class of Ricci semi-symmetric nearly Sasakian manifolds is a subclass of Einstein manifolds. [...] Read more.

A class of nearly Sasakian manifolds is considered in this paper. We discuss the geometric effects of some symmetries on such manifolds and show, under a certain condition, that the class of Ricci semi-symmetric nearly Sasakian manifolds is a subclass of Einstein manifolds. We prove that a Codazzi-type Ricci nearly Sasakian space form is either a Sasakian manifold with a constant

ϕ

-holomorphic sectional curvature

H = 1

or a 5-dimensional proper nearly Sasakian manifold with a constant

ϕ

-holomorphic sectional curvature

H > 1

. We also prove that the spectrum of the operator

H^{2}

generated by the nearly Sasakian space form is a set of a simple eigenvalue of 0 and an eigenvalue of multiplicity 4, and we induce that the underlying space form carries a Sasaki–Einstein structure. We show that there exist integrable distributions with totally geodesic leaves on the same manifolds, and we prove that there are no proper nearly Sasakian space forms with constant sectional curvature. Full article

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

9 pages, 266 KiB

Open AccessArticle

Nearly Sasakian Manifolds of Constant Type

by Aligadzhi Rustanov

Axioms 2022, 11(12), 673; https://doi.org/10.3390/axioms11120673 - 26 Nov 2022

Viewed by 1596

Abstract

The article deals with nearly Sasakian manifolds of a constant type. It is proved that the almost Hermitian structure induced on the integral manifolds of the maximum dimension of the first fundamental distribution of the nearly Sasakian manifold is a nearly Kähler manifold. [...] Read more.

The article deals with nearly Sasakian manifolds of a constant type. It is proved that the almost Hermitian structure induced on the integral manifolds of the maximum dimension of the first fundamental distribution of the nearly Sasakian manifold is a nearly Kähler manifold. It is proved that the class of nearly Sasakian manifolds of the zero constant type coincides with the class of Sasakian manifolds. The concept of constancy of the type of an almost contact metric manifold is introduced through its Nijenhuis tensor, and the criterion of constancy of the type of an almost contact metric manifold is proved. The coincidence of both concepts of type constancy for the nearly Sasakian manifold is proved. It is proved that the almost Hermitian structure induced on the integral manifolds of the maximum dimension of the first fundamental distribution of the almost contact metric manifold of the zero constant type is the Hermitian structure. Full article

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

24 pages, 352 KiB

Open AccessArticle

Geometry of Bi-Warped Product Submanifolds of Nearly Trans-Sasakian Manifolds

by Ali H. Alkhaldi and Akram Ali

Mathematics 2021, 9(8), 847; https://doi.org/10.3390/math9080847 - 13 Apr 2021

Cited by 2 | Viewed by 1864

Abstract

In the present work, we consider two types of bi-warped product submanifolds,

M = M_{T} \times_{f_{1}} M_{⊥} \times_{f_{2}} M_{ϕ}

and

M = M_{ϕ} \times_{f_{1}} M_{T} \times_{f_{2}} M_{⊥}

, in [...] Read more.

In the present work, we consider two types of bi-warped product submanifolds,

M = M_{T} \times_{f_{1}} M_{⊥} \times_{f_{2}} M_{ϕ}

and

M = M_{ϕ} \times_{f_{1}} M_{T} \times_{f_{2}} M_{⊥}

, in nearly trans-Sasakian manifolds and construct inequalities for the squared norm of the second fundamental form. The main results here are a generalization of several previous results. We also design some applications, in view of mathematical physics, and obtain relations between the second fundamental form and the Dirichlet energy. The relationship between the eigenvalues and the second fundamental form is also established. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

11 pages, 772 KiB

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 6 | Viewed by 2606

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

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