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12 pages, 288 KiB

Open AccessArticle

Characterizations of the Frame Bundle Admitting Metallic Structures on Almost Quadratic ϕ-Manifolds

by Mohammad Nazrul Islam Khan, Uday Chand De and Teg Alam

Mathematics 2023, 11(14), 3097; https://doi.org/10.3390/math11143097 - 13 Jul 2023

Viewed by 690

Abstract

In this work, we have characterized the frame bundle

F M

admitting metallic structures on almost quadratic

ϕ

-manifolds

ϕ^{2} = p ϕ + q I - q η \otimes ζ

, where p is an arbitrary constant and q is a [...] Read more.

In this work, we have characterized the frame bundle

F M

admitting metallic structures on almost quadratic

ϕ

-manifolds

ϕ^{2} = p ϕ + q I - q η \otimes ζ

, where p is an arbitrary constant and q is a nonzero constant. The complete lifts of an almost quadratic

ϕ

-structure to the metallic structure on

F M

are constructed. We also prove the existence of a metallic structure on

F M

with the aid of the

\tilde{J}

tensor field, which we define. Results for the 2-Form and its derivative are then obtained. Additionally, we derive the expressions of the Nijenhuis tensor of a tensor field

\tilde{J}

on

F M

. Finally, we construct an example of it to finish. Full article

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

9 pages, 246 KiB

Open AccessArticle

Solitonic View of Generic Contact CR-Submanifolds of Sasakian Manifolds with Concurrent Vector Fields

by Vandana, Rajeev Budhiraja, Aliya Naaz Siddiqui and Ali Hussain Alkhaldi

Mathematics 2023, 11(12), 2663; https://doi.org/10.3390/math11122663 - 12 Jun 2023

Cited by 1 | Viewed by 943

Abstract

This paper mainly devotes to the study of some solitons such as Ricci and Yamabe solitons and also their combination called Ricci-Yamabe solitons. In the geometry of solitons, a fundamental question is to identify the conditions under which these solitons can be trivial. [...] Read more.

This paper mainly devotes to the study of some solitons such as Ricci and Yamabe solitons and also their combination called Ricci-Yamabe solitons. In the geometry of solitons, a fundamental question is to identify the conditions under which these solitons can be trivial. Firstly, in this paper we study some extensive results on generic contact CR-submanifolds of Sasakian manifolds endowed with concurrent vector fields. Then some applications of solitons such as Ricci and Ricci-Yamabe solitons on such submanifolds with concurrent vector fields in the same ambient manifold have been discussed. Full article

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

10 pages, 263 KiB

Open AccessArticle

An Optimal Inequality for the Normal Scalar Curvature in Metallic Riemannian Space Forms

by Siraj Uddin, Majid Ali Choudhary and Najwa Mohammed Al-Asmari

Mathematics 2023, 11(10), 2252; https://doi.org/10.3390/math11102252 - 11 May 2023

Viewed by 782

Abstract

In this paper, we prove the DDVV conjecture for a slant submanifold in metallic Riemannian space forms with the semi-symmetric metric connection. The equality case of the derived inequality is discussed, and some special cases of the inequality are given. Full article

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

15 pages, 289 KiB

Open AccessArticle

Biharmonic Maps on f-Kenmotsu Manifolds with the Schouten–van Kampen Connection

by Hichem El hendi

Mathematics 2023, 11(8), 1905; https://doi.org/10.3390/math11081905 - 17 Apr 2023

Viewed by 912

Abstract

The object of the present paper was to study biharmonic maps on f-Kenmotsu manifolds and f-Kenmotsu manifolds with the Schouten–van Kampen connection. With the help of this connection, our results provided important insights related to harmonic and biharmonic maps. Full article

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

11 pages, 301 KiB

Open AccessArticle

Curvatures on Homogeneous Generalized Matsumoto Space

by M. K. Gupta, Suman Sharma, Fatemah Mofarreh and Sudhakar Kumar Chaubey

Mathematics 2023, 11(6), 1316; https://doi.org/10.3390/math11061316 - 9 Mar 2023

Cited by 1 | Viewed by 935

Abstract

The curvature characteristics of particular classes of Finsler spaces, such as homogeneous Finsler spaces, are one of the major issues in Finsler geometry. In this paper, we have obtained the expression for S-curvature in homogeneous Finsler space with a generalized Matsumoto metric [...] Read more.

The curvature characteristics of particular classes of Finsler spaces, such as homogeneous Finsler spaces, are one of the major issues in Finsler geometry. In this paper, we have obtained the expression for S-curvature in homogeneous Finsler space with a generalized Matsumoto metric and demonstrated that the homogeneous generalized Matsumoto space with isotropic S-curvature has to vanish the S-curvature. We have also derived the expression for the mean Berwald curvature by using the formula of S-curvature. Full article

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

18 pages, 319 KiB

Open AccessArticle

Geometry of Tangent Poisson–Lie Groups

by Ibrahim Al-Dayel, Foued Aloui and Sharief Deshmukh

Mathematics 2023, 11(1), 240; https://doi.org/10.3390/math11010240 - 3 Jan 2023

Viewed by 1161

Abstract

Let G be a Poisson–Lie group equipped with a left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on G to the tangent bundle

T G

of

G .

In this paper, we induce a left invariant contravariant [...] Read more.

Let G be a Poisson–Lie group equipped with a left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on G to the tangent bundle

T G

of

G .

In this paper, we induce a left invariant contravariant pseudo-Riemannian metric on the tangent bundle

T G

, and we express in different cases the contravariant Levi-Civita connection and curvature of

T G

in terms of the contravariant Levi-Civita connection and the curvature of G. We prove that the space of differential forms

Ω^{*} (G)

on G is a differential graded Poisson algebra if, and only if,

Ω^{*} (T G)

is a differential graded Poisson algebra. Moreover, we show that G is a pseudo-Riemannian Poisson–Lie group if, and only if, the Sanchez de Alvarez tangent Poisson–Lie group

T G

is also a pseudo-Riemannian Poisson–Lie group. Finally, some examples of pseudo-Riemannian tangent Poisson–Lie groups are given. Full article

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

12 pages, 259 KiB

Open AccessArticle

Lifts of a Quarter-Symmetric Metric Connection from a Sasakian Manifold to Its Tangent Bundle

by Mohammad Nazrul Islam Khan, Uday Chand De and Ljubica S. Velimirović

Mathematics 2023, 11(1), 53; https://doi.org/10.3390/math11010053 - 23 Dec 2022

Cited by 10 | Viewed by 1092

Abstract

The objective of this paper is to explore the complete lifts of a quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle. A relationship between the Riemannian connection and the quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle [...] Read more.

The objective of this paper is to explore the complete lifts of a quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle. A relationship between the Riemannian connection and the quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle was established. Some theorems on the curvature tensor and the projective curvature tensor of a Sasakian manifold with respect to the quarter-symmetric metric connection to its tangent bundle were proved. Finally, locally

ϕ

-symmetric Sasakian manifolds with respect to the quarter-symmetric metric connection to its tangent bundle were studied. Full article

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

8 pages, 270 KiB

Open AccessArticle

Locally Homogeneous Manifolds Defined by Lie Algebra of Infinitesimal Affine Transformations

by Vladimir A. Popov

Mathematics 2022, 10(24), 4654; https://doi.org/10.3390/math10244654 - 8 Dec 2022

Viewed by 677

Abstract

This article deals with Lie algebra

G

of all infinitesimal affine transformations of the manifold

M

with an affine connection, its stationary subalgebra

ℌ \subset G

, the Lie group

G

corresponding to the algebra

G

, and its subgroup

H \subset G

[...] Read more.

This article deals with Lie algebra

G

of all infinitesimal affine transformations of the manifold

M

with an affine connection, its stationary subalgebra

ℌ \subset G

, the Lie group

G

corresponding to the algebra

G

, and its subgroup

H \subset G

corresponding to the subalgebra

ℌ \subset G

. We consider the center

ℨ \subset G

and the commutant [

G, G]

of algebra

G

. The following condition for the closedness of the subgroup

H

in the group

G

is proved. If

ℌ \cap (ℨ + [G; G]) = ℌ \cap [G; G

], then

H

is closed in

G

. To prove it, an arbitrary group

G

is considered as a group of transformations of the set of left cosets

G / H

, where

H

is an arbitrary subgroup that does not contain normal subgroups of the group

G

. Among these transformations, we consider right multiplications. The group of right multiplications coincides with the center of the group

G

. However, it can contain the right multiplication by element

\bar{𝒽}

, belonging to normalizator of subgroup

H

and not belonging to the center of a group

G

. In the case when

G

is in the Lie group, corresponding to the algebra

G

of all infinitesimal affine transformations of the affine space

M

and its subgroup

H

corresponding to its stationary subalgebra

ℌ \subset G

, we prove that such element

\bar{𝒽}

exists if subgroup

H

is not closed in

G

. Moreover

\bar{𝒽}

belongs to the closures

\bar{H}

of subgroup

H

in

G

and does not belong to commutant

(G, G)

of group

G

. It is also proved that

H

is closed in

G

if

(P + ℨ) \cap ℌ = P \cap ℌ

for any semisimple algebra

P \in G

for which

P + ℜ = G

. Full article

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

13 pages, 318 KiB

Open AccessArticle

Some Inequalities of Hardy Type Related to Witten–Laplace Operator on Smooth Metric Measure Spaces

by Yanlin Li, Abimbola Abolarinwa, Ali H. Alkhaldi and Akram Ali

Mathematics 2022, 10(23), 4580; https://doi.org/10.3390/math10234580 - 2 Dec 2022

Cited by 33 | Viewed by 1444

Abstract

A complete Riemannian manifold equipped with some potential function and an invariant conformal measure is referred to as a complete smooth metric measure space. This paper generalizes some integral inequalities of the Hardy type to the setting of a complete non-compact smooth metric [...] Read more.

A complete Riemannian manifold equipped with some potential function and an invariant conformal measure is referred to as a complete smooth metric measure space. This paper generalizes some integral inequalities of the Hardy type to the setting of a complete non-compact smooth metric measure space without any geometric constraint on the potential function. The adopted approach highlights some criteria for a smooth metric measure space to admit Hardy inequalities related to Witten and Witten p-Laplace operators. The results in this paper complement in several aspect to those obtained recently in the non-compact setting. Full article

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

8 pages, 258 KiB

Open AccessArticle

Spheres and Tori as Elliptic Linear Weingarten Surfaces

by Dong-Soo Kim, Young Ho Kim and Jinhua Qian

Mathematics 2022, 10(21), 4065; https://doi.org/10.3390/math10214065 - 1 Nov 2022

Cited by 1 | Viewed by 818

Abstract

The linear Weingarten condition with ellipticity for the mean curvature and the extrinsic Gaussian curvature on a surface in the three-sphere can define a Riemannian metric which is called the elliptic linear Weingarten metric. We established some local characterizations of the round spheres [...] Read more.

The linear Weingarten condition with ellipticity for the mean curvature and the extrinsic Gaussian curvature on a surface in the three-sphere can define a Riemannian metric which is called the elliptic linear Weingarten metric. We established some local characterizations of the round spheres and the tori immersed in the 3-dimensional unit sphere, along with the Laplace operator, the spherical Gauss map and the Gauss map associated with the elliptic linear Weingarten metric. Full article

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

12 pages, 274 KiB

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 4 | Viewed by 1251

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)

15 pages, 301 KiB

Open AccessArticle

Representing Functions in H² on the Kepler Manifold via WPOAFD Based on the Rational Approximation of Holomorphic Functions

by Zeyuan Song and Zuoren Sun

Mathematics 2022, 10(15), 2729; https://doi.org/10.3390/math10152729 - 2 Aug 2022

Viewed by 1144

Abstract

The central problem of this study is to represent any holomorphic and square integrable function on the Kepler manifold in the series form based on Fourier analysis. Because these function spaces are reproducing kernel Hilbert spaces (RKHS), three different domains on the Kepler [...] Read more.

The central problem of this study is to represent any holomorphic and square integrable function on the Kepler manifold in the series form based on Fourier analysis. Because these function spaces are reproducing kernel Hilbert spaces (RKHS), three different domains on the Kepler manifold are considered and the weak pre-orthogonal adaptive Fourier decomposition (POAFD) is proposed on the domains. First, the weak maximal selection principle is shown to select the coefficient of the series. Furthermore, we prove the convergence theorem to show the accuracy of our method. This study is the extension of work by Wu et al. on POAFD in Bergman space. Full article

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

10 pages, 743 KiB

Open AccessArticle

A Discrete Representation of the Second Fundamental Form

by Alfonso Carriazo, Luis M. Fernández and Antonio Ramírez-de-Arellano

Mathematics 2022, 10(13), 2279; https://doi.org/10.3390/math10132279 - 29 Jun 2022

Viewed by 1194

Abstract

We present a new method to obtain a combinatorial representation for the behaviour of a submanifold isometrically immersed in a Riemannian manifold based on the second fundamental form. We also present several results applying this new way of representation. Full article

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

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Figure 1

10 pages, 263 KiB

Open AccessArticle

Generalized Wintgen Inequality for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature

by Aliya Naaz Siddiqui, Ali Hussain Alkhaldi and Lamia Saeed Alqahtani

Mathematics 2022, 10(10), 1727; https://doi.org/10.3390/math10101727 - 18 May 2022

Cited by 1 | Viewed by 963

Abstract

The geometry of Hessian manifolds is a fruitful branch of physics, statistics, Kaehlerian and affine differential geometry. The study of inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature was truly initiated in 2018 by Mihai, A. and Mihai, I. who [...] Read more.

The geometry of Hessian manifolds is a fruitful branch of physics, statistics, Kaehlerian and affine differential geometry. The study of inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature was truly initiated in 2018 by Mihai, A. and Mihai, I. who dealt with Chen-Ricci and Euler inequalities. Later on, Siddiqui, A.N., Ahmad K. and Ozel C. came with the study of Casorati inequality for statistical submanifolds in the same ambient space by using algebraic technique. Also, Chen, B.-Y., Mihai, A. and Mihai, I. obtained a Chen first inequality for such submanifolds. In 2020, Mihai, A. and Mihai, I. studied the Chen inequality for

δ (2, 2)

-invariant. In the development of this topic, we establish the generalized Wintgen inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Some examples are also discussed at the end. Full article

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

12 pages, 255 KiB

Open AccessArticle

On Statistical and Semi-Weyl Manifolds Admitting Torsion

by Adara M. Blaga and Antonella Nannicini

Mathematics 2022, 10(6), 990; https://doi.org/10.3390/math10060990 - 19 Mar 2022

Cited by 3 | Viewed by 1663

Abstract

We introduce the concept of quasi-semi-Weyl structure, we provide a couple of ways for constructing quasi-statistical and quasi-semi-Weyl structures by means of a pseudo-Riemannian metric, an affine connection and a tensor field on a smooth manifold, and we place these structures in relation [...] Read more.

We introduce the concept of quasi-semi-Weyl structure, we provide a couple of ways for constructing quasi-statistical and quasi-semi-Weyl structures by means of a pseudo-Riemannian metric, an affine connection and a tensor field on a smooth manifold, and we place these structures in relation with one another. Full article

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

12 pages, 268 KiB

Open AccessArticle

Yamabe Solitons on Some Conformal Almost Contact B-Metric Manifolds

by Mancho Manev

Mathematics 2022, 10(4), 658; https://doi.org/10.3390/math10040658 - 20 Feb 2022

Cited by 3 | Viewed by 1261

Abstract

A Yamabe soliton is defined on an arbitrary 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. The cases when the given manifold is cosymplectic or [...] Read more.

A Yamabe soliton is defined on an arbitrary 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. The cases when the given manifold is cosymplectic or Sasaki-like are studied. In this manner, manifolds are obtained that belong to one of the main classes of the studied manifolds. The same class contains the conformally equivalent manifolds of cosymplectic manifolds by the usual conformal transformation of the B-metric on contact distribution. In both cases, explicit five-dimensional examples are given, which are characterized in relation to the results obtained. Full article

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

10 pages, 272 KiB

Open AccessArticle

Para-Ricci-like Solitons with Arbitrary Potential on Para-Sasaki-like Riemannian Π-Manifolds

by Hristo Manev and Mancho Manev

Mathematics 2022, 10(4), 651; https://doi.org/10.3390/math10040651 - 19 Feb 2022

Cited by 1 | Viewed by 1015

Abstract

Para-Ricci-like solitons with arbitrary potential on para-Sasaki-like Riemannian

Π

-manifolds are introduced and studied. For the studied soliton, it is proved that its Ricci tensor is a constant multiple of the vertical component of both metrics. Thus, the corresponding scalar curvatures of both [...] Read more.

Para-Ricci-like solitons with arbitrary potential on para-Sasaki-like Riemannian

Π

-manifolds are introduced and studied. For the studied soliton, it is proved that its Ricci tensor is a constant multiple of the vertical component of both metrics. Thus, the corresponding scalar curvatures of both considered metrics are equal and constant. An explicit example of the Lie group as the manifold under study is presented. Full article

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

10 pages, 249 KiB

Open AccessArticle

Soliton-Type Equations on a Riemannian Manifold

by Nasser Bin Turki, Adara M. Blaga and Sharief Deshmukh

Mathematics 2022, 10(4), 633; https://doi.org/10.3390/math10040633 - 18 Feb 2022

Cited by 4 | Viewed by 1283

Abstract

We study some particular cases of soliton-type equations on a Riemannian manifold. We give an estimation of the first nonzero eigenvalue of the Laplace operator and provide necessary and sufficient conditions for the manifold to be isometric to a sphere. Finally, we characterize [...] Read more.

We study some particular cases of soliton-type equations on a Riemannian manifold. We give an estimation of the first nonzero eigenvalue of the Laplace operator and provide necessary and sufficient conditions for the manifold to be isometric to a sphere. Finally, we characterize trivial generalized gradient Ricci solitons. Full article

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

15 pages, 296 KiB

Open AccessArticle

Bounds for Statistical Curvatures of Submanifolds in Kenmotsu-like Statistical Manifolds

by Aliya Naaz Siddiqui, Mohd Danish Siddiqi and Ali Hussain Alkhaldi

Mathematics 2022, 10(2), 176; https://doi.org/10.3390/math10020176 - 6 Jan 2022

Cited by 4 | Viewed by 1053

Abstract

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized [...] Read more.

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized normalized Casorati curvatures. We also define the second fundamental form of those submanifolds that satisfy the equality condition. On Legendrian submanifolds of Kenmotsu-like statistical manifolds, we discuss a conjecture for Wintgen inequality. At the end, some immediate geometric consequences are stated. Full article

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

13 pages, 322 KiB

Open AccessArticle

f (R, T)

-Gravity Model with Perfect Fluid Admitting Einstein Solitons

by Mohd Danish Siddiqi, Sudhakar K. Chaubey and Mohammad Nazrul Islam Khan

Mathematics 2022, 10(1), 82; https://doi.org/10.3390/math10010082 - 27 Dec 2021

Cited by 13 | Viewed by 2148

Abstract

f (R, T)

-gravity is a generalization of Einstein’s field equations

({E F E}_{s})

and

f (R)

-gravity. In this research article, we demonstrate the virtues of the

f (R, T)

-gravity [...] Read more.

f (R, T)

-gravity is a generalization of Einstein’s field equations

({E F E}_{s})

and

f (R)

-gravity. In this research article, we demonstrate the virtues of the

f (R, T)

-gravity model with Einstein solitons

(E S)

and gradient Einstein solitons

(G E S)

. We acquire the equation of state of

f (R, T)

-gravity, provided the matter of

f (R, T)

-gravity is perfect fluid. In this series, we give a clue to determine pressure and density in radiation and phantom barrier era, respectively. It is proved that if a

f (R, T)

-gravity filled with perfect fluid admits an Einstein soliton

(g, ρ, λ)

and the Einstein soliton vector field

ρ

of

(g, ρ, λ)

is Killing, then the scalar curvature is constant and the Ricci tensor is proportional to the metric tensor. We also establish the Liouville’s equation in the

f (R, T)

-gravity model. Next, we prove that if a

f (R, T)

-gravity filled with perfect fluid admits a gradient Einstein soliton, then the potential function of gradient Einstein soliton satisfies Poisson equation. We also establish some physical properties of the

f (R, T)

-gravity model together with gradient Einstein soliton. Full article

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

13 pages, 567 KiB

Open AccessArticle

The Geometrical Characterizations of the Bertrand Curves of the Null Curves in Semi-Euclidean 4-Space

by Jianguo Sun and Yanping Zhao

Mathematics 2021, 9(24), 3294; https://doi.org/10.3390/math9243294 - 18 Dec 2021

Cited by 2 | Viewed by 2051

Abstract

According to the Frenet equations of the null curves in semi-Euclidean 4-space, the existence conditions and the geometrical characterizations of the Bertrand curves of the null curves are given in this paper. The examples and the graphs of the Bertrand pairs with two [...] Read more.

According to the Frenet equations of the null curves in semi-Euclidean 4-space, the existence conditions and the geometrical characterizations of the Bertrand curves of the null curves are given in this paper. The examples and the graphs of the Bertrand pairs with two different conditions are also given in order to supplement the conclusion of this paper more intuitively. Full article

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

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11 pages, 291 KiB

Open AccessArticle

η-∗-Ricci Solitons and Almost co-Kähler Manifolds

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

Mathematics 2021, 9(24), 3200; https://doi.org/10.3390/math9243200 - 11 Dec 2021

Cited by 8 | Viewed by 1795

Abstract

The subject of the present paper is the investigation of a new type of solitons, called

η

-∗-Ricci solitons in

(k, μ)

-almost co-Kähler manifold (briefly,

a c k m

), which generalizes the notion of the

η

-Ricci soliton [...] Read more.

The subject of the present paper is the investigation of a new type of solitons, called

η

-∗-Ricci solitons in

(k, μ)

-almost co-Kähler manifold (briefly,

a c k m

), which generalizes the notion of the

η

-Ricci soliton introduced by Cho and Kimura. First, the expression of the ∗-Ricci tensor on

a c k m

is obtained. Additionally, we classify the

η

-∗-Ricci solitons in

(k, μ)

-

a c k m

s. Next, we investigate

(k, μ)

-

a c k m

s admitting gradient

η

-∗-Ricci solitons. Finally, we construct two examples to illustrate our results. Full article

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

9 pages, 269 KiB

Open AccessArticle

On Minimal Hypersurfaces of a Unit Sphere

by Amira Ishan, Sharief Deshmukh, Ibrahim Al-Dayel and Cihan Özgür

Mathematics 2021, 9(24), 3161; https://doi.org/10.3390/math9243161 - 8 Dec 2021

Viewed by 1844

Abstract

Minimal compact hypersurface in the unit sphere

S^{n + 1}

having squared length of shape operator

{∥A∥}^{2} < n

are totally geodesic and with

{∥A∥}^{2} = n

are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance [...] Read more.

Minimal compact hypersurface in the unit sphere

S^{n + 1}

having squared length of shape operator

{∥A∥}^{2} < n

are totally geodesic and with

{∥A∥}^{2} = n

are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in

S^{n + 1}

. One finds a naturally induced vector field w called the associated vector field and a smooth function

ρ

called support function on the hypersurface M of

S^{n + 1}

. It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in

S^{5}

to be totally geodesic is that the support function

ρ

is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in

S^{n + 1}

, (

n > 2

), provided the scalar curvature

τ

is a constant on integral curves of w. Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in

S^{n + 1}

is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field

A w

. Full article

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

Review

Jump to: Research

38 pages, 518 KiB

Open AccessFeature PaperReview

Differential Geometry of Submanifolds in Complex Space Forms Involving δ-Invariants

by Bang-Yen Chen, Adara M. Blaga and Gabriel-Eduard Vîlcu

Mathematics 2022, 10(4), 591; https://doi.org/10.3390/math10040591 - 14 Feb 2022

Cited by 9 | Viewed by 2014

Abstract

One of the fundamental problems in the theory of submanifolds is to establish optimal relationships between intrinsic and extrinsic invariants for submanifolds. In order to establish such relations, the first author introduced in the 1990s the notion of

δ

-invariants for Riemannian manifolds, [...] Read more.

One of the fundamental problems in the theory of submanifolds is to establish optimal relationships between intrinsic and extrinsic invariants for submanifolds. In order to establish such relations, the first author introduced in the 1990s the notion of

δ

-invariants for Riemannian manifolds, which are different in nature from the classical curvature invariants. The earlier results on

δ

-invariants and their applications have been summarized in the first author’s book published in 2011 Pseudo-Riemannian Geometry, δ-Invariants and Applications (ISBN: 978-981-4329-63-7). In this survey, we present a comprehensive account of the development of the differential geometry of submanifolds in complex space forms involving the

δ

-invariants done mostly after the publication of the book. Full article

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

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