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11 pages, 265 KiB  
Article
Pair of Associated η-Ricci–Bourguignon Almost Solitons with Vertical Potential on Sasaki-like Almost Contact Complex Riemannian Manifolds
by Mancho Manev
Mathematics 2025, 13(11), 1863; https://doi.org/10.3390/math13111863 - 3 Jun 2025
Cited by 1 | Viewed by 375
Abstract
The manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism acts as [...] Read more.
The manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism acts as an anti-isometry for these metrics, called B-metrics, if its action is restricted to the contact distribution of the manifold. In this paper, some curvature properties of a special class of these manifolds, called Sasaki-like, are studied. Such a manifold is defined by the condition that its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). Each of the two B-metrics on the considered manifold is specialized here as an η-Ricci–Bourguignon almost soliton, where η is the contact form, i.e., has an additional curvature property such that the metric is a self-similar solution of a special intrinsic geometric flow. Almost solitons are generalizations of solitons because their defining condition uses functions rather than constants as coefficients. The introduced (almost) solitons are a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein). The soliton potential is chosen to be collinear with the Reeb vector field and is therefore called vertical. The special case of the soliton potential being solenoidal (i.e., divergence-free) with respect to each of the B-metrics is also considered. The resulting manifolds equipped with the pair of associated η-Ricci–Bourguignon almost solitons are characterized geometrically. An example of arbitrary dimension is constructed and the properties obtained in the theoretical part are confirmed. Full article
(This article belongs to the Special Issue Analysis on Differentiable Manifolds)
15 pages, 325 KiB  
Article
η-Ricci Solitons on Weak β-Kenmotsu f-Manifolds
by Vladimir Rovenski
Mathematics 2025, 13(11), 1734; https://doi.org/10.3390/math13111734 - 24 May 2025
Viewed by 245
Abstract
Recent interest among geometers in f-structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by the [...] Read more.
Recent interest among geometers in f-structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by the author and R. Wolak as a generalization of Hermitian and Kähler structures, as well as f-structures, allow for a fresh perspective on the classical theory. In this paper, we study a new f-structure of this kind, called the weak β-Kenmotsu f-structure, as a generalization of K. Kenmotsu’s concept. We prove that a weak β-Kenmotsu f-manifold is a locally twisted product of the Euclidean space and a weak Kähler manifold. Our main results show that such manifolds with β=const and equipped with an η-Ricci soliton structure whose potential vector field satisfies certain conditions are η-Einstein manifolds of constant scalar curvature. Full article
(This article belongs to the Special Issue Differential Geometric Structures and Their Applications)
15 pages, 251 KiB  
Article
Solitons on Submanifolds of Kenmotsu Manifolds with Concurrent Vector Fields
by Vandana, Meraj Ali Khan and Aliya Naaz Siddiqui
Symmetry 2025, 17(4), 500; https://doi.org/10.3390/sym17040500 - 26 Mar 2025
Viewed by 337
Abstract
The present research paper investigates submanifolds of Kenmotsu manifolds, focusing on those equipped with concurrent vector fields. It examines the structural and geometric properties of such submanifolds, analyzing the decomposed equations in both vertical and horizontal components. Furthermore, the study generalizes certain results [...] Read more.
The present research paper investigates submanifolds of Kenmotsu manifolds, focusing on those equipped with concurrent vector fields. It examines the structural and geometric properties of such submanifolds, analyzing the decomposed equations in both vertical and horizontal components. Furthermore, the study generalizes certain results in the context of η-Ricci solitons and η-Yamabe solitons. Full article
(This article belongs to the Section Mathematics)
24 pages, 395 KiB  
Review
Geometry of Weak Metric f-Manifolds: A Survey
by Vladimir Rovenski
Mathematics 2025, 13(4), 556; https://doi.org/10.3390/math13040556 - 8 Feb 2025
Cited by 1 | Viewed by 563
Abstract
The interest of geometers in f-structures is motivated by the study of the dynamics of contact foliations, as well as their applications in physics. A weak f-structure on a smooth manifold, introduced by the author and R. Wolak, generalizes K. Yano’s [...] Read more.
The interest of geometers in f-structures is motivated by the study of the dynamics of contact foliations, as well as their applications in physics. A weak f-structure on a smooth manifold, introduced by the author and R. Wolak, generalizes K. Yano’s f-structure. This generalization allows us to revisit classical theory and discover applications of Killing vector fields, totally geodesic foliations, Ricci-type solitons, and Einstein-type metrics. This article reviews the results regarding weak metric f-manifolds and their distinguished classes. Full article
(This article belongs to the Special Issue Recent Studies in Differential Geometry and Its Applications)
23 pages, 619 KiB  
Article
A System of Tensor Equations over the Dual Split Quaternion Algebra with an Application
by Liuqing Yang, Qing-Wen Wang and Zuliang Kou
Mathematics 2024, 12(22), 3571; https://doi.org/10.3390/math12223571 - 15 Nov 2024
Cited by 4 | Viewed by 913
Abstract
In this paper, we propose a definition of block tensors and the real representation of tensors. Equipped with the simplification method, i.e., the real representation along with the M-P inverse, we demonstrate the conditions that are necessary and sufficient for the system of [...] Read more.
In this paper, we propose a definition of block tensors and the real representation of tensors. Equipped with the simplification method, i.e., the real representation along with the M-P inverse, we demonstrate the conditions that are necessary and sufficient for the system of dual split quaternion tensor equations (ANX,XSC)=(B,D), when its solution exists. Furthermore, the general expression of the solution is also provided when the solution of the system exists, and we use a numerical example to validate it in the last section. To the best of our knowledge, this is the first time that the aforementioned tensor system has been examined on dual split quaternion algebra. Additionally, we provide its equivalent conditions when its Hermitian solution X=X and η-Hermitian solutions X=Xη exist. Subsequently, we discuss two special dual split quaternion tensor equations. Last but not least, we propose an application for encrypting and decrypting two color videos, and we validate this algorithm through a specific example. Full article
(This article belongs to the Special Issue Advances of Linear and Multilinear Algebra)
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28 pages, 407 KiB  
Article
Continuity Equation of Transverse Kähler Metrics on Sasakian Manifolds
by Yushuang Fan and Tao Zheng
Mathematics 2024, 12(19), 3132; https://doi.org/10.3390/math12193132 - 7 Oct 2024
Viewed by 899
Abstract
We introduce the continuity equation of transverse Kähler metrics on Sasakian manifolds and establish its interval of maximal existence. When the first basic Chern class is null (resp. negative), we prove that the solution of the (resp. normalized) continuity equation converges smoothly to [...] Read more.
We introduce the continuity equation of transverse Kähler metrics on Sasakian manifolds and establish its interval of maximal existence. When the first basic Chern class is null (resp. negative), we prove that the solution of the (resp. normalized) continuity equation converges smoothly to the unique η-Einstein metric in the basic Bott–Chern cohomological class of the initial transverse Kähler metric (resp. first basic Chern class). These results are the transverse version of the continuity equation of the Kähler metrics studied by La Nave and Tian, and also counterparts of the Sasaki–Ricci flow studied by Smoczyk, Wang, and Zhang. Full article
(This article belongs to the Special Issue Complex Analysis and Geometric Function Theory, 2nd Edition)
15 pages, 326 KiB  
Article
Solitons of η-Ricci–Bourguignon Type on Submanifolds in (LCS)m Manifolds
by Lixu Yan, Vandana, Aliya Naaz Siddiqui, Halil Ibrahim Yoldas and Yanlin Li
Symmetry 2024, 16(6), 675; https://doi.org/10.3390/sym16060675 - 31 May 2024
Viewed by 1006
Abstract
In this research article, we concentrate on the exploration of submanifolds in an (LCS)m-manifold B˜. We examine these submanifolds in the context of two distinct vector fields, namely, the characteristic vector field and the concurrent [...] Read more.
In this research article, we concentrate on the exploration of submanifolds in an (LCS)m-manifold B˜. We examine these submanifolds in the context of two distinct vector fields, namely, the characteristic vector field and the concurrent vector field. Initially, we consider some classifications of η-Ricci–Bourguignon (in short, η-RB) solitons on both invariant and anti-invariant submanifolds of B˜ employing the characteristic vector field. We establish several significant findings through this process. Furthermore, we investigate additional results by using η-RB solitons on invariant submanifolds of B˜ with concurrent vector fields, and discuss a supporting example. Full article
12 pages, 766 KiB  
Article
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 1704
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)
31 pages, 1293 KiB  
Article
Study of a Minimally Deformed Anisotropic Solution for Compact Objects with Massive Scalar Field in Brans–Dicke Gravity Admitting the Karmarkar Condition
by M. K. Jasim, Ksh. Newton Singh, Abdelghani Errehymy, S. K. Maurya and M. V. Mandke
Universe 2023, 9(5), 208; https://doi.org/10.3390/universe9050208 - 26 Apr 2023
Cited by 5 | Viewed by 1536
Abstract
In the present paper, we focused on exploring the possibility of providing a new class of exact solutions for viable anisotropic stellar systems by means of the massive Brans–Dicke (BD) theory of gravity. In this respect, we used the decoupling of gravitational sources [...] Read more.
In the present paper, we focused on exploring the possibility of providing a new class of exact solutions for viable anisotropic stellar systems by means of the massive Brans–Dicke (BD) theory of gravity. In this respect, we used the decoupling of gravitational sources by minimal geometric deformation (MGD) (eη=Ψ+βh) for compact stellar objects in the realm of embedding class-one space-time to study anisotropic solutions for matter sources through the modified Einstein field equations. For this purpose, we used the ansatz for Ψ relating to the prominent, well-known and well-behaved Finch–Skea model via Karmarkar condition, and the determination scheme for deformation function h(r) was proposed via mimic requirement on radial pressure component: θ11(r)=pr(r) and matter density: θ00(r)=ρ(r) for the anisotropic sector. Moreover, we analyzed the main physical highlights of the anisotropic celestial object by executing several physical tests for the case θ11(r)=pr(r). We have clearly shown how the parameters α, β and ωBD introduced by massive BD gravity via the MGD approach incorporating the anisotropic profile of the matter distribution have an immense effect on many physical parameters of compact bodies such as LMC X-4, LMC X-4, Her X-1, 4U 1820-30, 4U 1608-52, SAX J1808.4–658 and many others that can be fitted. Full article
(This article belongs to the Section Cosmology)
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11 pages, 267 KiB  
Article
On Bochner Flat Kähler B-Manifolds
by Cornelia-Livia Bejan, Galia Nakova and Adara M. Blaga
Axioms 2023, 12(4), 336; https://doi.org/10.3390/axioms12040336 - 30 Mar 2023
Cited by 2 | Viewed by 1826
Abstract
We obtain on a Kähler B-manifold (i.e., a Kähler manifold with a Norden metric) some corresponding results from the Kählerian and para-Kählerian context concerning the Bochner curvature. We prove that such a manifold is of constant totally real sectional curvatures if and only [...] Read more.
We obtain on a Kähler B-manifold (i.e., a Kähler manifold with a Norden metric) some corresponding results from the Kählerian and para-Kählerian context concerning the Bochner curvature. We prove that such a manifold is of constant totally real sectional curvatures if and only if it is a holomorphic Einstein, Bochner flat manifold. Moreover, we provide the necessary and sufficient conditions for a gradient Ricci soliton or a holomorphic η-Einstein Kähler manifold with a Norden metric to be Bochner flat. Finally, we show that a Kähler B-manifold is of quasi-constant totally real sectional curvatures if and only if it is a holomorphic η-Einstein, Bochner flat manifold. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application)
14 pages, 309 KiB  
Article
Certain Curvature Conditions on Kenmotsu Manifolds and ★-η-Ricci Solitons
by Halil İbrahim Yoldaş, Abdul Haseeb and Fatemah Mofarreh
Axioms 2023, 12(2), 140; https://doi.org/10.3390/axioms12020140 - 30 Jan 2023
Cited by 11 | Viewed by 2125
Abstract
The present paper deals with the investigations of a Kenmotsu manifold satisfying certain curvature conditions endowed with -η-Ricci solitons. First we find some necessary conditions for such a manifold to be φ-Einstein. Then, we study the notion of [...] Read more.
The present paper deals with the investigations of a Kenmotsu manifold satisfying certain curvature conditions endowed with -η-Ricci solitons. First we find some necessary conditions for such a manifold to be φ-Einstein. Then, we study the notion of -η-Ricci soliton on this manifold and prove some significant results related to this notion. Finally, we construct a nontrivial example of three-dimensional Kenmotsu manifolds to verify some of our results. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application)
14 pages, 342 KiB  
Article
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 1749
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)
16 pages, 351 KiB  
Article
Conformal η-Ricci Solitons on Riemannian Submersions under Canonical Variation
by Mohd. Danish Siddiqi, Ali Hussain Alkhaldi, Meraj Ali Khan and Aliya Naaz Siddiqui
Axioms 2022, 11(11), 594; https://doi.org/10.3390/axioms11110594 - 27 Oct 2022
Cited by 6 | Viewed by 1690
Abstract
This research article endeavors to discuss the attributes of Riemannian submersions under the canonical variation in terms of the conformal η-Ricci soliton and gradient conformal η-Ricci soliton with a potential vector field ζ. Additionally, we estimate the various conditions for [...] Read more.
This research article endeavors to discuss the attributes of Riemannian submersions under the canonical variation in terms of the conformal η-Ricci soliton and gradient conformal η-Ricci soliton with a potential vector field ζ. Additionally, we estimate the various conditions for which the target manifold of Riemannian submersion under the canonical variation is a conformal η-Ricci soliton with a Killing vector field and a φ(Ric)-vector field. Moreover, we deduce the generalized Liouville equation for Riemannian submersion under the canonical variation satisfying by a last multiplier Ψ of the vertical potential vector field ζ and show that the base manifold of Riemanian submersion under canonical variation is an η Einstein for gradient conformal η-Ricci soliton with a scalar concircular field γ on base manifold. Finally, we illustrate an example of Riemannian submersions between Riemannian manifolds, which verify our results. Full article
13 pages, 334 KiB  
Article
Geometry of Indefinite Kenmotsu Manifolds as *η-Ricci-Yamabe Solitons
by Abdul Haseeb, Mohd Bilal, Sudhakar K. Chaubey and Mohammad Nazrul Islam Khan
Axioms 2022, 11(9), 461; https://doi.org/10.3390/axioms11090461 - 7 Sep 2022
Cited by 12 | Viewed by 1934
Abstract
In this paper, we study the properties of ϵ-Kenmotsu manifolds if its metrics are *η-Ricci-Yamabe solitons. It is proven that an ϵ-Kenmotsu manifold endowed with a *η-Ricci-Yamabe soliton is η-Einstein. The necessary conditions for an ϵ [...] Read more.
In this paper, we study the properties of ϵ-Kenmotsu manifolds if its metrics are *η-Ricci-Yamabe solitons. It is proven that an ϵ-Kenmotsu manifold endowed with a *η-Ricci-Yamabe soliton is η-Einstein. The necessary conditions for an ϵ-Kenmotsu manifold, whose metric is a *η-Ricci-Yamabe soliton, to be an Einstein manifold are derived. Finally, we model an indefinite Kenmotsu manifold example of dimension 5 to examine the existence *η-Ricci-Yamabe solitons. Full article
(This article belongs to the Section Geometry and Topology)
33 pages, 455 KiB  
Review
Energy-Momentum Complex in Higher Order Curvature-Based Local Gravity
by Salvatore Capozziello, Maurizio Capriolo and Gaetano Lambiase
Particles 2022, 5(3), 298-330; https://doi.org/10.3390/particles5030026 - 10 Aug 2022
Cited by 7 | Viewed by 2386
Abstract
An unambiguous definition of gravitational energy remains one of the unresolved issues of physics today. This problem is related to the non-localization of gravitational energy density. In General Relativity, there have been many proposals for defining the gravitational energy density, notably those proposed [...] Read more.
An unambiguous definition of gravitational energy remains one of the unresolved issues of physics today. This problem is related to the non-localization of gravitational energy density. In General Relativity, there have been many proposals for defining the gravitational energy density, notably those proposed by Einstein, Tolman, Landau and Lifshitz, Papapetrou, Møller, and Weinberg. In this review, we firstly explored the energy–momentum complex in an nth order gravitational Lagrangian L=Lgμν,gμν,i1,gμν,i1i2,gμν,i1i2i3,,gμν,i1i2i3in and then in a gravitational Lagrangian as Lg=(R¯+a0R2+k=1pakRkR)g. Its gravitational part was obtained by invariance of gravitational action under infinitesimal rigid translations using Noether’s theorem. We also showed that this tensor, in general, is not a covariant object but only an affine object, that is, a pseudo-tensor. Therefore, the pseudo-tensor ταη becomes the one introduced by Einstein if we limit ourselves to General Relativity and its extended corrections have been explicitly indicated. The same method was used to derive the energy–momentum complex in fR gravity both in Palatini and metric approaches. Moreover, in the weak field approximation the pseudo-tensor ταη to lowest order in the metric perturbation h was calculated. As a practical application, the power per unit solid angle Ω emitted by a localized source carried by a gravitational wave in a direction x^ for a fixed wave number k under a suitable gauge was obtained, through the average value of the pseudo-tensor over a suitable spacetime domain and the local conservation of the pseudo-tensor. As a cosmological application, in a flat Friedmann–Lemaître–Robertson–Walker spacetime, the gravitational and matter energy density in f(R) gravity both in Palatini and metric formalism was proposed. The gravitational energy–momentum pseudo-tensor could be a useful tool to investigate further modes of gravitational radiation beyond two standard modes required by General Relativity and to deal with non-local theories of gravity involving k terms. Full article
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