Mathematics

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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

Viewed by 447

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)

12 pages, 265 KiB

Open AccessArticle

Chern Flat and Chern Ricci-Flat Twisted Product Hermitian Manifolds

by Shuwen Li, Yong He, Weina Lu and Ruijia Yang

Mathematics 2024, 12(3), 449; https://doi.org/10.3390/math12030449 - 30 Jan 2024

Viewed by 602

Abstract

Let

(M_{1}, g)

and

(M_{2}, h)

be two Hermitian manifolds. The twisted product Hermitian manifold

(M_{1} \times M_{2}^{f}, G)

is the product manifold

M_{1} \times M_{2}

endowed [...] Read more.

Let

(M_{1}, g)

and

(M_{2}, h)

be two Hermitian manifolds. The twisted product Hermitian manifold

(M_{1} \times M_{2}^{f}, G)

is the product manifold

M_{1} \times M_{2}

endowed with the Hermitian metric

G = g + f^{2} h

, where f is a positive smooth function on

M_{1} \times M_{2}

. In this paper, the Chern curvature, Chern Ricci curvature, Chern Ricci scalar curvature and holomorphic sectional curvature of the twisted product Hermitian manifold are derived. The necessary and sufficient conditions for the compact twisted product Hermitian manifold to have constant holomorphic sectional curvature are obtained. Under the condition that the logarithm of the twisted function is pluriharmonic, it is proved that the twisted product Hermitian manifold is Chern flat or Chern Ricci-flat, if and only if

(M_{1}, g)

and

(M_{2}, h)

are Chern flat or Chern Ricci-flat, respectively. Full article

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

0 pages, 345 KiB

Open AccessArticle

An Invariant of Riemannian Type for Legendrian Warped Product Submanifolds of Sasakian Space Forms

by Fatemah Abdullah Alghamdi, Lamia Saeed Alqahtani, Ali H. Alkhaldi and Akram Ali

Mathematics 2023, 11(23), 4718; https://doi.org/10.3390/math11234718 - 21 Nov 2023

Viewed by 554

Abstract

In the present paper, we investigate the geometry and topology of warped product Legendrian submanifolds in Sasakian space forms

D^{2 n + 1} (ϵ)

and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the [...] Read more.

In the present paper, we investigate the geometry and topology of warped product Legendrian submanifolds in Sasakian space forms

D^{2 n + 1} (ϵ)

and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the length of the warping functions. This inequality also involves intrinsic invariants (

δ

-invariant and sectional curvature). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the gradient Ricci curvature. Some new results on mean curvature vanishing are presented as a partial solution to the well-known problem given by S.S. Chern. Full article

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

16 pages, 323 KiB

Open AccessArticle

Metallic Structures for Tangent Bundles over Almost Quadratic ϕ-Manifolds

by Mohammad Nazrul Islam Khan, Sudhakar Kumar Chaubey, Nahid Fatima and Afifah Al Eid

Mathematics 2023, 11(22), 4683; https://doi.org/10.3390/math11224683 - 17 Nov 2023

Viewed by 667

Abstract

This paper aims to explore the metallic structure

J^{2} = p J + q I,

where p and q are natural numbers, using complete and horizontal lifts on the tangent bundle

T M

over almost quadratic

ϕ

-structures (briefly, [...] Read more.

This paper aims to explore the metallic structure

J^{2} = p J + q I,

where p and q are natural numbers, using complete and horizontal lifts on the tangent bundle

T M

over almost quadratic

ϕ

-structures (briefly,

(ϕ, ξ, η)

). Tensor fields

\tilde{F}

and

F^{*}

are defined on

T M

, and it is shown that they are metallic structures over

(ϕ, ξ, η)

. Next, the fundamental 2-form

Ω

and its derivative

d Ω

, with the help of complete lift on

T M

over

(ϕ, ξ, η)

, are evaluated. Furthermore, the integrability conditions and expressions of the Lie derivative of metallic structures

\tilde{F}

and

F^{*}

are determined using complete and horizontal lifts on

T M

over

(ϕ, ξ, η)

, respectively. Finally, we prove the existence of almost quadratic

ϕ

-structures on

T M

with non-trivial examples. Full article

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

15 pages, 300 KiB

Open AccessArticle

Tangent Bundles Endowed with Quarter-Symmetric Non-Metric Connection (QSNMC) in a Lorentzian Para-Sasakian Manifold

by Rajesh Kumar, Lalnunenga Colney, Samesh Shenawy and Nasser Bin Turki

Mathematics 2023, 11(19), 4163; https://doi.org/10.3390/math11194163 - 04 Oct 2023

Cited by 2 | Viewed by 667

Abstract

The purpose of the present paper is to study the complete lifts of a QSNMC from an LP-Sasakian manifold to its tangent bundle. The lifts of the curvature tensor, Ricci tensor, projective Ricci tensor, and lifts of Einstein manifold endowed with QSNMC in [...] Read more.

The purpose of the present paper is to study the complete lifts of a QSNMC from an LP-Sasakian manifold to its tangent bundle. The lifts of the curvature tensor, Ricci tensor, projective Ricci tensor, and lifts of Einstein manifold endowed with QSNMC in an LP-Sasakian manifold to its tangent bundle are investigated. Necessary and sufficient conditions for the lifts of the Ricci tensor to be symmetric and skew-symmetric and the lifts of the projective Ricci tensor to be skew-symmetric in the tangent bundle are given. An example of complete lifts of four-dimensional LP-Sasakian manifolds in the tangent bundle is shown. Full article

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

14 pages, 3569 KiB

Open AccessArticle

Framed Natural Mates of Framed Curves in Euclidean 3-Space

by Yanlin Li and Mahmut Mak

Mathematics 2023, 11(16), 3571; https://doi.org/10.3390/math11163571 - 17 Aug 2023

Cited by 15 | Viewed by 1140

Abstract

In this study, we consider framed curves as regular or singular space curves with an adapted frame in Euclidean 3-space. We define framed natural mates of a framed curve that are tangent to the generalized principal normal of the framed curve. Subsequently, we [...] Read more.

In this study, we consider framed curves as regular or singular space curves with an adapted frame in Euclidean 3-space. We define framed natural mates of a framed curve that are tangent to the generalized principal normal of the framed curve. Subsequently, we present the relationships between a framed curve and its framed natural mates. In particular, we establish some necessary and sufficient conditions for the framed natural mates of specific framed curves, such as framed spherical curves, framed helices, framed slant helices, and framed rectifying curves. Finally, we support the concept with some examples. Full article

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

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12 pages, 1065 KiB

Open AccessArticle

Surface Pencil Pair Interpolating Bertrand Pair as Common Asymptotic Curves in Euclidean 3-Space

by Fatemah Mofarreh and Rashad A. Abdel-Baky

Mathematics 2023, 11(16), 3495; https://doi.org/10.3390/math11163495 - 13 Aug 2023

Cited by 2 | Viewed by 690

Abstract

In this paper, we obtain the necessary and sufficient conditions of a surface pencil pair interpolating a Bertrand pair as common asymptotic curves in Euclidean 3-space

E^{3}

. Afterwards, the conclusion to the ruled surface pencil pair is also obtained. Meanwhile, the [...] Read more.

In this paper, we obtain the necessary and sufficient conditions of a surface pencil pair interpolating a Bertrand pair as common asymptotic curves in Euclidean 3-space

E^{3}

. Afterwards, the conclusion to the ruled surface pencil pair is also obtained. Meanwhile, the epitomes are stated to emphasize that the proposed methods are effective in product manufacturing by adjusting the shapes of the surface pencil pair. Full article

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

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13 pages, 293 KiB

Open AccessArticle

The Homology of Warped Product Submanifolds of Spheres and Their Applications

by Lamia Saeed Alqahtani, Akram Ali, Pişcoran Laurian-Ioan and Ali H. Alkhaldi

Mathematics 2023, 11(15), 3405; https://doi.org/10.3390/math11153405 - 04 Aug 2023

Cited by 1 | Viewed by 553

Abstract

The aim of the current article is to formulate sufficient conditions for the Laplacian and a gradient of the warping function of a compact warped product submanifold

Σ^{β_{1} + β_{2}}

in a unit sphere

S^{d}

that provides trivial homology [...] Read more.

The aim of the current article is to formulate sufficient conditions for the Laplacian and a gradient of the warping function of a compact warped product submanifold

Σ^{β_{1} + β_{2}}

in a unit sphere

S^{d}

that provides trivial homology and fundamental groups. We also validate the instability of current flows in

π_{1} (Σ^{β_{1} + β_{2}})

. The constraints are also applied to the warped function eigenvalues and integral Ricci curvatures. Full article

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

8 pages, 258 KiB

Open AccessArticle

Conformal Transformations on General (α,β)-Spaces

by Xiaoling Zhang, Xuesong Zhang and Mengke Wu

Mathematics 2023, 11(15), 3381; https://doi.org/10.3390/math11153381 - 02 Aug 2023

Viewed by 536

Abstract

In this paper, we study conformal transformations between two almost regular general

(α, β)

-metrics. By using the method of special coordinate system, the necessary and sufficient conditions for conformal transformations preserving the mean Landsberg curvature are obtained. Further, a [...] Read more.

In this paper, we study conformal transformations between two almost regular general

(α, β)

-metrics. By using the method of special coordinate system, the necessary and sufficient conditions for conformal transformations preserving the mean Landsberg curvature are obtained. Further, a rigidity theorem for regular general

(α, β)

-metrics is proved. Full article

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

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 - 09 Jun 2023

Cited by 1 | Viewed by 804

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)

11 pages, 306 KiB

Open AccessArticle

Geometry of CR-Slant Warped Products in Nearly Kaehler Manifolds

by Siraj Uddin, Bang-Yen Chen and Rawan Bossly

Mathematics 2023, 11(12), 2600; https://doi.org/10.3390/math11122600 - 07 Jun 2023

Viewed by 768

Abstract

Recently, we studied CR-slant warped products

B_{1} \times_{f} M_{⊥}

, where

B_{1} = M_{T} \times M_{θ}

is the Riemannian product of holomorphic and proper slant submanifolds and

M_{⊥}

is a totally real submanifold in a nearly [...] Read more.

Recently, we studied CR-slant warped products

B_{1} \times_{f} M_{⊥}

, where

B_{1} = M_{T} \times M_{θ}

is the Riemannian product of holomorphic and proper slant submanifolds and

M_{⊥}

is a totally real submanifold in a nearly Kaehler manifold. In the continuation, in this paper, we study

B_{2} \times_{f} M_{θ}

, where

B_{2} = M_{T} \times M_{⊥}

is a CR-product of a nearly Kaehler manifold and establish Chen’s inequality for the squared norm of the second fundamental form. Some special cases of Chen’s inequality are given. Full article

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

12 pages, 766 KiB

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 - 08 May 2023

Viewed by 1004

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)

15 pages, 320 KiB

Open AccessArticle

The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres

by Ibrahim Al-Dayel and Sharief Deshmukh

Mathematics 2023, 11(8), 1958; https://doi.org/10.3390/math11081958 - 21 Apr 2023

Viewed by 902

Abstract

We studied the random variable

V_{t} = {vol}_{S^{2}} (g_{t} B \cap B)

, where B is a disc on the sphere

S^{2}

centered at the north pole and

{(g_{t})}_{t \geq 0}

is [...] Read more.

We studied the random variable

V_{t} = {vol}_{S^{2}} (g_{t} B \cap B)

, where B is a disc on the sphere

S^{2}

centered at the north pole and

{(g_{t})}_{t \geq 0}

is the Brownian motion on the special orthogonal group

S O (3)

starting at the identity. We applied the results of the theory of compact Lie groups to evaluate the expectation of

V_{t}

for

0 \leq t \leq τ

, where

τ

is the first time when

V_{t}

vanishes. We obtained an integral formula using the heat equation on some Riemannian submanifold

Γ_{B}

seen as the support of the function

f (g) = {vol}_{S^{2}} (g B \cap B)

immersed in

S O (3)

. The integral formula depends on the mean curvature of

Γ_{B}

and the diameter of B. Full article

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

13 pages, 253 KiB

Open AccessArticle

A Note on the Geometry of RW Space-Times

by Sameh Shenawy, Uday Chand De and Nasser Bin Turki

Mathematics 2023, 11(6), 1440; https://doi.org/10.3390/math11061440 - 16 Mar 2023

Cited by 1 | Viewed by 1078

Abstract

A conformally flat GRW space-time is a perfect fluid RW space-time. In this note, we investigated the influence of many differential curvature conditions, such as the existence of recurrent and semi-symmetric curvature tensors. In each case, the form of the Ricci curvature tensor, [...] Read more.

A conformally flat GRW space-time is a perfect fluid RW space-time. In this note, we investigated the influence of many differential curvature conditions, such as the existence of recurrent and semi-symmetric curvature tensors. In each case, the form of the Ricci curvature tensor, the energy–momentum tensor, the energy density, the pressure of the fluid, and the equation of state are determined and interpreted. For example, it is demonstrated that a Ricci semi-symmetric RW space-time reduces to Einstein space-time or a Ricci recurrent RW space-time, and the perfect fluid space-time is referred to as Yang pure space-time or dark matter era. Full article

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

15 pages, 312 KiB

Open AccessArticle

Li–Yau-Type Gradient Estimate along Geometric Flow

by Shyamal Kumar Hui, Abimbola Abolarinwa, Meraj Ali Khan, Fatemah Mofarreh, Apurba Saha and Sujit Bhattacharyya

Mathematics 2023, 11(6), 1364; https://doi.org/10.3390/math11061364 - 10 Mar 2023

Cited by 2 | Viewed by 1161

Abstract

In this article we derive a Li–Yau-type gradient estimate for a generalized weighted parabolic heat equation with potential on a weighted Riemannian manifold evolving by a geometric flow. As an application, a Harnack-type inequality is also derived in the end. Full article

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

9 pages, 285 KiB

Open AccessArticle

The λ-Point Map between Two Legendre Plane Curves

by Azeb Alghanemi and Abeer AlGhawazi

Mathematics 2023, 11(4), 997; https://doi.org/10.3390/math11040997 - 15 Feb 2023

Cited by 1 | Viewed by 850

Abstract

The

λ

-point map between two Legendre plane curves, which is a map from the plane into the plane, is introduced. The singularity of this map is studied through this paper and many known plane map singularities are realized as special cases of [...] Read more.

The

λ

-point map between two Legendre plane curves, which is a map from the plane into the plane, is introduced. The singularity of this map is studied through this paper and many known plane map singularities are realized as special cases of this construction. Precisely, the corank one and corank two singularities of the

λ

-point map between two Legendre plane curves are investigated and the geometric conditions for this map to have corank one singularities, such as fold, cusp, swallowtail, lips, and beaks are obtained. Additionally, the geometric conditions for the

λ

-point map to have a sharksfin singularity, which is a corank two singularity, are obtained. 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 4 | Viewed by 1139

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)

18 pages, 343 KiB

Open AccessArticle

Optimal Inequalities for Hemi-Slant Riemannian Submersions

by Mehmet Akif Akyol, Ramazan Demir, Nergiz Önen Poyraz and Gabriel-Eduard Vîlcu

Mathematics 2022, 10(21), 3993; https://doi.org/10.3390/math10213993 - 27 Oct 2022

Cited by 1 | Viewed by 1044

Abstract

In the present paper, we establish some basic inequalities involving the Ricci and scalar curvature of the vertical and the horizontal distributions for hemi-slant submersions having the total space a complex space form. We also discuss the equality case of the obtained inequalities [...] Read more.

In the present paper, we establish some basic inequalities involving the Ricci and scalar curvature of the vertical and the horizontal distributions for hemi-slant submersions having the total space a complex space form. We also discuss the equality case of the obtained inequalities and provide illustrative examples. Full article

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

11 pages, 287 KiB

Open AccessArticle

Application of Mixed Generalized Quasi-Einstein Spacetimes in General Relativity

by Mohd Vasiulla, Abdul Haseeb, Fatemah Mofarreh and Mohabbat Ali

Mathematics 2022, 10(20), 3749; https://doi.org/10.3390/math10203749 - 12 Oct 2022

Cited by 3 | Viewed by 1076

Abstract

In the present article, some geometric and physical properties of

M G {(Q E)}_{n}

were investigated. Moreover, general relativistic viscous fluid

M G {(Q E)}_{4}

spacetimes with some physical applications were studied. Finally, through a non-trivial example [...] Read more.

In the present article, some geometric and physical properties of

M G {(Q E)}_{n}

were investigated. Moreover, general relativistic viscous fluid

M G {(Q E)}_{4}

spacetimes with some physical applications were studied. Finally, through a non-trivial example of

M G {(Q E)}_{4}

spacetime, we proved its existence. Full article

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

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