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18 pages, 331 KB

Open AccessArticle

The Generalized Symmetric Non-Metric Connection and Its Applications to ∗-Conformal η-Ricci–Yamabe Solitons on α-Cosymplectic Manifolds

by Laltluangkima Chawngthu, Rajesh Kumar, Oğuzhan Bahadır and Md Aquib

Axioms 2025, 14(12), 858; https://doi.org/10.3390/axioms14120858 - 23 Nov 2025

Viewed by 407

Abstract

This paper investigates the geometric properties of ∗-conformal

η

-Ricci–Yamabe solitons (∗-conformal

η

-RYS) on

α

-cosymplectic manifolds (

α - CSM

) equipped with a newly introduced connection known as the generalized symmetric non-metric connection (GSNMC). The existence of this connection is [...] Read more.

This paper investigates the geometric properties of ∗-conformal

η

-Ricci–Yamabe solitons (∗-conformal

η

-RYS) on

α

-cosymplectic manifolds (

α - CSM

) equipped with a newly introduced connection known as the generalized symmetric non-metric connection (GSNMC). The existence of this connection is rigorously established, and a thorough analysis is conducted on various curvature characteristics of

α - CSM

manifolds in the context of the GSNMC. This paper further explores the behavior, classification, and properties of ∗-conformal

η

-RYS, including their applications in different geometric settings. A particular focus is placed on the harmonic interpretation of ∗-conformal

η

-RYS associated with the GSNMC on

α - CSM

. To substantiate the theoretical developments, an explicit example is provided: a five-dimensional

α

-cosymplectic metric is constructed that incorporates a ∗-conformal

η

-RYS structure with respect to the proposed connection. This example serves to illustrate the practical applicability of the results and validates the theoretical framework presented in the paper. Full article

(This article belongs to the Special Issue Recent Developments in Differential Geometry and Its Applications)

14 pages, 290 KB

Open AccessArticle

Z-Solitons and Gradient Z-Solitons on α-Cosymplectic Manifolds

by Mustafa Yildirim, Mehmet Akif Akyol, Majid Ali Choudhary and Foued Aloui

Axioms 2025, 14(10), 759; https://doi.org/10.3390/axioms14100759 - 10 Oct 2025

Viewed by 541

Abstract

In this paper, we study Z-solitons and gradient Z-solitons on

α

-cosymplectic manifolds. The soliton structure is defined by the generalized tensor

Z = S + β g

, where S denotes the Ricci tensor, g the metric tensor, and

β

[...] Read more.

In this paper, we study Z-solitons and gradient Z-solitons on

α

-cosymplectic manifolds. The soliton structure is defined by the generalized tensor

Z = S + β g

, where S denotes the Ricci tensor, g the metric tensor, and

β

a smooth function. We investigate the geometric implications of Z-solitons under various curvature conditions, with a focus on the interplay between the Z-tensor and the Q-curvature tensor, as well as the case of Z-recurrent

α

-cosymplectic manifolds. Our classification results establish that such manifolds can be Einstein,

η

-Einstein, or of constant curvature. Finally, we construct a concrete five-dimensional example of an

α

-cosymplectic manifold that admits a Z-soliton structure, thereby illustrating the theoretical framework. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 3rd Edition)

6 pages, 177 KB

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 1 | Viewed by 757

Abstract

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)

16 pages, 290 KB

Open AccessArticle

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 1183

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)

17 pages, 339 KB

Open AccessArticle

Three-Dimensional Semi-Symmetric Almost α-Cosymplectic Manifolds

by Sermin Öztürk and Hakan Öztürk

Symmetry 2023, 15(11), 2022; https://doi.org/10.3390/sym15112022 - 5 Nov 2023

Viewed by 1336

Abstract

The main objective of this paper is to study semi-symmetric almost

α

-cosymplectic three-manifolds. We present basic formulas for almost

α

-cosymplectic manifolds. Using curvature properties, we obtain some necessary and sufficient conditions on semi-symmetric almost

α

-cosymplectic three-manifolds. We obtain the main [...] Read more.

The main objective of this paper is to study semi-symmetric almost

α

-cosymplectic three-manifolds. We present basic formulas for almost

α

-cosymplectic manifolds. Using curvature properties, we obtain some necessary and sufficient conditions on semi-symmetric almost

α

-cosymplectic three-manifolds. We obtain the main results under an additional condition. The paper concludes with two illustrative examples. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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 1866

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 1472

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 1365

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)

12 pages, 766 KB

Open AccessArticle

Quarter-Symmetric Metric Connection on a Cosymplectic Manifold

by Miroslav D. Maksimović and Milan Lj. Zlatanović

Mathematics 2023, 11(9), 2209; https://doi.org/10.3390/math11092209 - 8 May 2023

Cited by 1 | Viewed by 1905

Abstract

We study the quarter-symmetric metric A-connection on a cosymplectic manifold. Observing linearly independent curvature tensors with respect to the quarter-symmetric metric A-connection, we construct the Weyl projective curvature tensor on a cosymplectic manifold. In this way, we obtain new conditions for [...] Read more.

We study the quarter-symmetric metric A-connection on a cosymplectic manifold. Observing linearly independent curvature tensors with respect to the quarter-symmetric metric A-connection, we construct the Weyl projective curvature tensor on a cosymplectic manifold. In this way, we obtain new conditions for the manifold to be projectively flat. At the end of the paper, we define

η

-Einstein cosymplectic manifolds of the

θ

-th kind and prove that they coincide with the

η

-Einstein cosymplectic manifold. Full article

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

10 pages, 282 KB

Open AccessArticle

Yamabe Solitons on Conformal Almost-Contact Complex Riemannian Manifolds with Vertical Torse-Forming Vector Field

by Mancho Manev

Axioms 2023, 12(1), 44; https://doi.org/10.3390/axioms12010044 - 1 Jan 2023

Cited by 2 | Viewed by 2250

Abstract

A Yamabe soliton is considered on an almost-contact complex Riemannian manifold (also known as an almost-contact B-metric manifold), which is obtained by a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. A case [...] Read more.

A Yamabe soliton is considered on an almost-contact complex Riemannian manifold (also known as an almost-contact B-metric manifold), which is obtained by a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. A case in which the potential is a torse-forming vector field of constant length on the vertical distribution determined by the Reeb vector field is studied. In this way, manifolds from one of the main classes of the studied manifolds are obtained. The same class contains the conformally equivalent manifolds of cosymplectic manifolds by the usual conformal transformation of the given B-metric. An explicit five-dimensional example of a Lie group is given, which is characterized in relation to the obtained results. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Geometry and Topology)

14 pages, 342 KB

Open AccessFeature PaperArticle

Almost Riemann Solitons with Vertical Potential on Conformal Cosymplectic Contact Complex Riemannian Manifolds

by Mancho Manev

Symmetry 2023, 15(1), 104; https://doi.org/10.3390/sym15010104 - 30 Dec 2022

Cited by 2 | Viewed by 1956

Abstract

Almost-Riemann solitons are introduced and studied on an almost contact complex Riemannian manifold, i.e., an almost-contact B-metric manifold, which is obtained from a cosymplectic manifold of the considered type by means of a contact conformal transformation of the Reeb vector field, its dual [...] Read more.

Almost-Riemann solitons are introduced and studied on an almost contact complex Riemannian manifold, i.e., an almost-contact B-metric manifold, which is obtained from a cosymplectic manifold of the considered type by means of a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. The potential of the studied soliton is assumed to be in the vertical distribution, i.e., it is collinear to the Reeb vector field. In this way, manifolds from the four main classes of the studied manifolds are obtained. The curvature properties of the resulting manifolds are derived. An explicit example of dimension five is constructed. The Bochner curvature tensor is used (for a dimension of at least seven) as a conformal invariant to obtain these properties and to construct an explicit example in relation to the obtained results. Full article

(This article belongs to the Special Issue Symmetry and Geometry in Physics II)

20 pages, 309 KB

Open AccessArticle

From Dual Connections to Almost Contact Structures

by Emmanuel Gnandi and Stéphane Puechmorel

Mathematics 2022, 10(20), 3822; https://doi.org/10.3390/math10203822 - 16 Oct 2022

Viewed by 2275

Abstract

A dualistic structure on a smooth Riemaniann manifold M is a triple

(M, g, \nabla)

with g a Riemaniann metric and ∇ an affine connection generally assumed to be torsionless. From g and ∇, dual connection

\nabla^{*}

can [...] Read more.

A dualistic structure on a smooth Riemaniann manifold M is a triple

(M, g, \nabla)

with g a Riemaniann metric and ∇ an affine connection generally assumed to be torsionless. From g and ∇, dual connection

\nabla^{*}

can be defined. In this work, we give conditions on the basis of this notion for a manifold to admit an almost contact structure and some related structures: almost contact metric, contact, contact metric, cosymplectic, and co-Kähler in the three-dimensional case. Full article

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

17 pages, 316 KB

Open AccessArticle

A Study of Clairaut Semi-Invariant Riemannian Maps from Cosymplectic Manifolds

by Yanlin Li, Rajendra Prasad, Abdul Haseeb, Sushil Kumar and Sumeet Kumar

Axioms 2022, 11(10), 503; https://doi.org/10.3390/axioms11100503 - 26 Sep 2022

Cited by 23 | Viewed by 2419

Abstract

In the present note, we characterize Clairaut semi-invariant Riemannian maps from cosymplectic manifolds to Riemannian manifolds. Moreover, we provide a nontrivial example of such a Riemannian map. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application)

12 pages, 274 KB

Open AccessArticle

h-Almost Ricci–Yamabe Solitons in Paracontact Geometry

by Uday Chand De, Mohammad Nazrul Islam Khan and Arpan Sardar

Mathematics 2022, 10(18), 3388; https://doi.org/10.3390/math10183388 - 18 Sep 2022

Cited by 7 | Viewed by 2126

Abstract

In this article, we classify h-almost Ricci–Yamabe solitons in paracontact geometry. In particular, we characterize para-Kenmotsu manifolds satisfying h-almost Ricci–Yamabe solitons and 3-dimensional para-Kenmotsu manifolds obeying h-almost gradient Ricci–Yamabe solitons. Then, we classify para-Sasakian manifolds and para-cosymplectic manifolds admitting h [...] Read more.

In this article, we classify h-almost Ricci–Yamabe solitons in paracontact geometry. In particular, we characterize para-Kenmotsu manifolds satisfying h-almost Ricci–Yamabe solitons and 3-dimensional para-Kenmotsu manifolds obeying h-almost gradient Ricci–Yamabe solitons. Then, we classify para-Sasakian manifolds and para-cosymplectic manifolds admitting h-almost Ricci–Yamabe solitons and h-almost gradient Ricci–Yamabe solitons, respectively. Finally, we construct an example to illustrate our result. Full article

(This article belongs to the Special Issue Geometry of Manifolds and Applications)

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 4 | Viewed by 3033

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)

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