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15 pages, 295 KiB

Open AccessArticle

k-Almost Newton-Conformal Ricci Solitons on Hypersurfaces Within Golden Riemannian Manifolds with Constant Golden Sectional Curvature

by Amit Kumar Rai, Majid Ali Choudhary, Mohd. Danish Siddiqi, Ghodratallah Fasihi-Ramandi, Uday Chand De and Ion Mihai

Axioms 2025, 14(8), 579; https://doi.org/10.3390/axioms14080579 - 26 Jul 2025

Viewed by 248

Abstract

The current work establishes the geometrical bearing for hypersurfaces in a Golden Riemannian manifold with constant golden sectional curvature with respect to k-almost Newton-conformal Ricci solitons. Moreover, we extensively explore the immersed r-almost Newton-conformal Ricci soliton and determine the sufficient conditions [...] Read more.

The current work establishes the geometrical bearing for hypersurfaces in a Golden Riemannian manifold with constant golden sectional curvature with respect to k-almost Newton-conformal Ricci solitons. Moreover, we extensively explore the immersed r-almost Newton-conformal Ricci soliton and determine the sufficient conditions for total geodesicity with adequate restrictions on some smooth functions using mathematical operators. Furthermore, we go over some natural conclusions in which the gradient k-almost Newton-conformal Ricci soliton on the hypersurface of the Golden Riemannian manifold becomes compact. Finally, we establish a Schur’s type inequality in terms of k-almost Newton-conformal Ricci solitons immersed in Golden Riemannian manifolds with constant golden sectional curvature. Full article

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

12 pages, 233 KiB

Open AccessArticle

Rigidity Characterizations of Conformal Solitons

by Junsheng Gong and Jiancheng Liu

Mathematics 2025, 13(11), 1837; https://doi.org/10.3390/math13111837 - 31 May 2025

Viewed by 303

Abstract

We study the rigidity of conformal solitons, give a sufficient and necessary condition that guarantees that every closed conformal soliton is gradient conformal soliton, and prove that complete conformal solitons with a nonpositive Ricci curvature must be trivial under an integral condition. In [...] Read more.

We study the rigidity of conformal solitons, give a sufficient and necessary condition that guarantees that every closed conformal soliton is gradient conformal soliton, and prove that complete conformal solitons with a nonpositive Ricci curvature must be trivial under an integral condition. In particular, by using a p-harmonic map from a complete gradient conformal soliton in a smooth Riemannian manifold, we classify complete noncompact nontrivial gradient conformal solitons under some suitable conditions, and similar results are given for gradient Yamabe solitons and gradient k-Yamabe solitons. Full article

(This article belongs to the Section B: Geometry and Topology)

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 428

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, 284 KiB

Open AccessArticle

Exploring Conformal Soliton Structures in Tangent Bundles with Ricci-Quarter Symmetric Metric Connections

by Yanlin Li, Aydin Gezer and Erkan Karakas

Mathematics 2024, 12(13), 2101; https://doi.org/10.3390/math12132101 - 4 Jul 2024

Cited by 10 | Viewed by 964

Abstract

In this study, we investigate the tangent bundle

T M

of an n-dimensional (pseudo-)Riemannian manifold M equipped with a Ricci-quarter symmetric metric connection

\tilde{\nabla}

. Our primary goal is to establish the necessary and sufficient conditions for

T M

to exhibit [...] Read more.

In this study, we investigate the tangent bundle

T M

of an n-dimensional (pseudo-)Riemannian manifold M equipped with a Ricci-quarter symmetric metric connection

\tilde{\nabla}

. Our primary goal is to establish the necessary and sufficient conditions for

T M

to exhibit characteristics of various solitons, specifically conformal Yamabe solitons, gradient conformal Yamabe solitons, conformal Ricci solitons, and gradient conformal Ricci solitons. We determine that for

T M

to be a conformal Yamabe soliton, the potential vector field must satisfy certain conditions when lifted vertically, horizontally, or completely from M to

T M

, alongside specific constraints on the conformal factor

λ

and the geometric properties of M. For gradient conformal Yamabe solitons, the conditions involve

λ

and the Hessian of the potential function. Similarly, for

T M

to be a conformal Ricci soliton, we identify conditions involving the lift of the potential vector field, the value of

λ

, and the curvature properties of M. For gradient conformal Ricci solitons, the criteria include the Hessian of the potential function and the Ricci curvature of M. These results enhance the understanding of the geometric properties of tangent bundles under Ricci-quarter symmetric metric connections and provide insights into their transition into various soliton states, contributing significantly to the field of differential geometry. Full article

(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications)

18 pages, 293 KiB

Open AccessArticle

Analyzing Curvature Properties and Geometric Solitons of the Twisted Sasaki Metric on the Tangent Bundle over a Statistical Manifold

by Lixu Yan, Yanlin Li, Lokman Bilen and Aydın Gezer

Mathematics 2024, 12(9), 1395; https://doi.org/10.3390/math12091395 - 2 May 2024

Cited by 3 | Viewed by 1463

Abstract

Let

(M, \nabla, g)

be a statistical manifold and

T M

be its tangent bundle endowed with a twisted Sasaki metric G. This paper serves two primary objectives. The first objective is to investigate the curvature properties of [...] Read more.

Let

(M, \nabla, g)

be a statistical manifold and

T M

be its tangent bundle endowed with a twisted Sasaki metric G. This paper serves two primary objectives. The first objective is to investigate the curvature properties of the tangent bundle

T M

. The second objective is to explore conformal vector fields and Ricci, Yamabe, and gradient Ricci–Yamabe solitons on the tangent bundle

T M

according to the twisted Sasaki metric G. Full article

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

8 pages, 230 KiB

Open AccessArticle

Gradient Ricci Solitons on Spacelike Hypersurfaces of Lorentzian Manifolds Admitting a Closed Conformal Timelike Vector Field

by Norah Alshehri and Mohammed Guediri

Mathematics 2024, 12(6), 842; https://doi.org/10.3390/math12060842 - 13 Mar 2024

Cited by 1 | Viewed by 1023

Abstract

In this article, we investigate Ricci solitons occurring on spacelike hypersurfaces of Einstein Lorentzian manifolds. We give the necessary and sufficient conditions for a spacelike hypersurface of a Lorentzian manifold, equipped with a closed conformal timelike vector field

\bar{ξ}

, to be [...] Read more.

In this article, we investigate Ricci solitons occurring on spacelike hypersurfaces of Einstein Lorentzian manifolds. We give the necessary and sufficient conditions for a spacelike hypersurface of a Lorentzian manifold, equipped with a closed conformal timelike vector field

\bar{ξ}

, to be a gradient Ricci soliton having its potential function as the inner product of

\bar{ξ}

and the timelike unit normal vector field to the hypersurface. Moreover, when the ambient manifold is Einstein and the hypersurface is compact, we establish that, under certain straightforward conditions, the hypersurface is an extrinsic sphere, that is, a totally umbilical hypersurface with a non-zero constant mean curvature. In particular, if the ambient Lorentzian manifold has a constant sectional curvature, we show that the compact spacelike hypersurface is essentially a round sphere. Full article

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

14 pages, 315 KiB

Open AccessArticle

ζ-Conformally Flat LP-Kenmotsu Manifolds and Ricci–Yamabe Solitons

by Abdul Haseeb, Mohd Bilal, Sudhakar K. Chaubey and Abdullah Ali H. Ahmadini

Mathematics 2023, 11(1), 212; https://doi.org/10.3390/math11010212 - 31 Dec 2022

Cited by 12 | Viewed by 1840

Abstract

In the present paper, we characterize m-dimensional

ζ

-conformally flat

L P

-Kenmotsu manifolds (briefly,

{(L P K)}_{m}

) equipped with the Ricci–Yamabe solitons (RYS) and gradient Ricci–Yamabe solitons (GRYS). It is proven that the scalar curvature r of [...] Read more.

In the present paper, we characterize m-dimensional

ζ

-conformally flat

L P

-Kenmotsu manifolds (briefly,

{(L P K)}_{m}

) equipped with the Ricci–Yamabe solitons (RYS) and gradient Ricci–Yamabe solitons (GRYS). It is proven that the scalar curvature r of an

{(L P K)}_{m}

admitting an RYS satisfies the Poisson equation

Δ r = \frac{4 (m - 1)}{δ} {β (m - 1) + ρ} + 2 (m - 3) r - 4 m (m - 1) (m - 2)

, where

ρ, δ (\neq 0) \in R

. In this sequel, the condition for which the scalar curvature of an

{(L P K)}_{m}

admitting an RYS holds the Laplace equation is established. We also give an affirmative answer for the existence of a GRYS on an

{(L P K)}_{m}

. Finally, a non-trivial example of an

L P

-Kenmotsu manifold

(L P K)

of dimension four is constructed to verify some of our results. Full article

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

16 pages, 351 KiB

Open AccessArticle

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 1691

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 (R i c)

-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, 298 KiB

Open AccessArticle

Geometrical Structure in a Perfect Fluid Spacetime with Conformal Ricci–Yamabe Soliton

by Pengfei Zhang, Yanlin Li, Soumendu Roy, Santu Dey and Arindam Bhattacharyya

Symmetry 2022, 14(3), 594; https://doi.org/10.3390/sym14030594 - 17 Mar 2022

Cited by 28 | Viewed by 2564

Abstract

The present paper aims to deliberate the geometric composition of a perfect fluid spacetime with torse-forming vector field

ξ

in connection with conformal Ricci–Yamabe metric and conformal

η

-Ricci–Yamabe metric. We delineate the conditions for conformal Ricci–Yamabe soliton to be expanding, steady or [...] Read more.

The present paper aims to deliberate the geometric composition of a perfect fluid spacetime with torse-forming vector field

ξ

in connection with conformal Ricci–Yamabe metric and conformal

η

-Ricci–Yamabe metric. We delineate the conditions for conformal Ricci–Yamabe soliton to be expanding, steady or shrinking. We also discuss conformal Ricci–Yamabe soliton on some special types of perfect fluid spacetime such as dust fluid, dark fluid and radiation era. Furthermore, we design conformal

η

-Ricci–Yamabe soliton to find its characteristics in a perfect fluid spacetime and lastly acquired Laplace equation from conformal

η

-Ricci–Yamabe soliton equation when the potential vector field

ξ

of the soliton is of gradient type. Overall, the main novelty of the paper is to study the geometrical phenomena and characteristics of our newly introduced conformal Ricci–Yamabe and conformal

η

-Ricci–Yamabe solitons to apply their existence in a perfect fluid spacetime. Full article

(This article belongs to the Section Mathematics)

16 pages, 329 KiB

Open AccessArticle

Geometry of α-Cosymplectic Metric as ∗-Conformal η-Ricci–Yamabe Solitons Admitting Quarter-Symmetric Metric Connection

by Pengfei Zhang, Yanlin Li, Soumendu Roy and Santu Dey

Symmetry 2021, 13(11), 2189; https://doi.org/10.3390/sym13112189 - 16 Nov 2021

Cited by 9 | Viewed by 2046

Abstract

The outline of this research article is to initiate the development of a ∗-conformal

η

-Ricci–Yamabe soliton in

α

-Cosymplectic manifolds according to the quarter-symmetric metric connection. Here, we have established some curvature properties of

α

-Cosymplectic manifolds in regard to the quarter-symmetric [...] Read more.

The outline of this research article is to initiate the development of a ∗-conformal

η

-Ricci–Yamabe soliton in

α

-Cosymplectic manifolds according to the quarter-symmetric metric connection. Here, we have established some curvature properties of

α

-Cosymplectic manifolds in regard to the quarter-symmetric metric connection. Further, the attributes of the soliton when the manifold gratifies a quarter-symmetric metric connection have been displayed in this article. Later, we picked up the Laplace equation from ∗-conformal

η

-Ricci–Yamabe soliton equation when the potential vector field

ξ

of the soliton is of gradient type, admitting quarter-symmetric metric connection. Next, we evolved the nature of the soliton when the vector field’s conformal killing reveals a quarter-symmetric metric connection. We show an example of a 5-dimensional

α

-cosymplectic metric as a ∗-conformal

η

-Ricci–Yamabe soliton acknowledges quarter-symmetric metric connection to prove our results. Full article

13 pages, 292 KiB

Open AccessArticle

Conformally Flat Siklos Metrics Are Ricci Solitons

by Giovanni Calvaruso

Axioms 2020, 9(2), 64; https://doi.org/10.3390/axioms9020064 - 8 Jun 2020

Cited by 7 | Viewed by 2450

Abstract

We study and solve the Ricci soliton equation for an arbitrary locally conformally flat Siklos metric, proving that such spacetimes are always Ricci solitons. Full article

(This article belongs to the Special Issue Pseudo-Riemannian Metrics and Applications)

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