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Geometry of Submanifolds and Applications

herausgegeben von: Bang-Yen Chen, Majid Ali Choudhary, Mohammad Nazrul Islam Khan

Verlag: Springer Nature Singapore

Buchreihe : Infosys Science Foundation Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book features chapters written by renowned scientists from various parts of the world, providing an up-to-date survey of submanifold theory, spanning diverse topics and applications. The book covers a wide range of topics such as Chen–Ricci inequalities in differential geometry, optimal inequalities for Casorati curvatures in quaternion geometry, conformal η-Ricci–Yamabe solitons, submersion on statistical metallic structure, solitons in f(R, T)-gravity, metric-affine geometry, generalized Wintgen inequalities, tangent bundles, and Lagrangian submanifolds.
Moreover, the book showcases the latest findings on Pythagorean submanifolds and submanifolds of four-dimensional f-manifolds. The chapters in this book delve into numerous problems and conjectures on submanifolds, providing valuable insights for scientists, educators, and graduate students looking to stay updated with the latest developments in the field. With its comprehensive coverage and detailed explanations, this book is an essential resource for anyone interested in submanifold theory.

Inhaltsverzeichnis

Frontmatter

Recent Developments on Chen–Ricci Inequalities in Differential Geometry

Abstract

One of the most fundamental interests in submanifold theory is to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of submanifolds and find their applications. For this respect, the first author established in 1996 a basic inequality involving the Ricci curvature and the squared mean curvature of submanifolds in real space forms, which is known today as the Chen–Ricci inequality. Since then, there have been many papers dealing with this inequality. The purpose of this article is thus to present a comprehensive survey on recent developments in this inequality done by many geometers during the last 25 years.

Bang-Yen Chen, Adara M. Blaga

Solitons in -Gravity

Abstract

The goal of this book chapter is to examine the perfect fluid spacetimes fulfilling \(f(\mathcal {R},T)\)-gravity when Ricci and gradient Ricci solitons, Yamabe and gradient Yamabe solitons, \(\eta \)-Ricci and gradient \(\eta \)-Ricci solitons are its metrics. We establish criteria for which Ricci solitons are steady, expanding, or shrinking. Moreover, we study gradient Ricci solitons and prove that either the perfect fluid spacetime represents the dark energy era, or the spacetime has zero vorticity under certain conditions. Moreover, we investigate perfect fluid spacetimes fulfilling \(f(\mathcal {R},T)\)-gravity permitting Yamabe solitons and gradient Yamabe solitons and show that the integral curves of the velocity vector field \(\rho \) are geodesics and energy density and isotropic pressure remain invariant under \(\rho \) and for the gradient case the Ricci scalar is constant. Also, we study \(\eta \)-Ricci and gradient \(\eta \)-Ricci solitons in \(f(\mathcal {R},T)\)-gravity, respectively. Specifically, we establish criteria in which \(\eta \)-Ricci solitons are shrinking, expanding, or steady and for gradient \(\eta \)-Ricci solitons, either the spacetime represents the equation of state \(p+\sigma =\) constant, or the perfect fluid has vanishing vorticity.

Uday Chand De, Krishnendu De

A Survey on Lagrangian Submanifolds of Nearly Kaehler Six-Sphere

Abstract

We first review the nearly Kaehler structure on the six-dimensional unit sphere \(S^6\) and its Lagrangian (totally real) submanifolds. Then we present a survey of results on Lagrangian submanifolds M of the nearly Kähler \(S^6\) in terms of a canonically induced almost contact metric structure, Chen’s equality, normal connection, normal curvature operator, Ricci tensor and conformal flatness. In particular, conditions for M to be Sasakian and totally geodesic unit three-sphere are presented.

Ramesh Sharma

Pythagorean Submanifolds

Abstract

In this paper, we study a particular class of submanifolds, which we call Pythagorean submanifolds, in one of the standard complete simply connected models of real space forms. They are defined by an equation based on the shape operator. We give several examples and observe that any Pythagorean submanifold is isoparametric where the principal curvatures are given in terms of the Golden ratio. We also classify Pythagorean hypersurfaces.

Muhittin Evren Aydın, Adela Mihai, Cihan Özgür

On 4-Dimensional Submanifolds of f-Manifolds

Abstract

The present paper is devoted to a class of manifolds which admit an f-structure with 2-dimensional parallelizable kernel. Such manifolds are called 4-dimensional almost bi-contact metric manifolds; they carry a locally conformal almost Kähler structure. We give some classifications and prove their fundamental properties, then we deduce some properties about the complex manifolds associated with them. We show the existence of such manifolds by giving some non-trivial examples. Finally, we establish an interesting class and construct a concrete example.

Gherici Beldjilali

Almost Yamabe Solitons on a Total Space of Almost Hermitian Submersions

Abstract

This article presents the study of almost Hermitian submersion from an almost Yamabe soliton onto an almost Hermitian manifold. A non-trivial example is also mentioned in order to guarantee the existence of such solitons on the total space of almost Hermitian submersions. We mainly focus on Kaehler submersions from Kaehler manifolds which are a special case of almost Hermitian submersions. Under certain conditions, we find out that the fibres and the base manifold of such submersions are almost Yamabe solitons. We give the characterizations for an almost Yamabe soliton of a Kaehler submersion to be shrinking, steady and expanding in terms of extrinsic horizontal scalar curvature. Moreover, we observe the behaviour of torqued, recurrent and concurrent vector fields of the total space of the Kaehler submersion. In particular, we obtain characterization for an almost Yamabe soliton consisting of concurrent vector fields. Meanwhile, we give some results of such submersions when the total space is a Yamabe soliton which is a particular case of an almost Yamabe soliton.

Tanveer Fatima, Mehmet Akif Akyol, Rakesh Kumar

Generalized Wintgen Inequalities for -Para Sasakian Manifold

Abstract

This study explores generalized Wintgen inequalities for a contact pseudo-slant submanifold of \((\epsilon )\)-para Sasakian space form. Further, we describe the submanifolds where equality scenarios are valid and present several applications of the main finding. Additionally, we create an inequality for Ricci solitons to discover connections between intrinsic and extrinsic invariants.

Majid Ali Choudhary, Lovejoy S. Das, Mohd. Danish Siddiqi, Oğuzhan Bahadır

Certain Optimal Inequalities for Casorati Curvatures in Quaternion Geometry

Abstract

In this book chapter, we compute two inequalities for generalized normalized \(\delta \)-Casorati curvatures of quaternion bi-slant submanifolds in quaternion space forms. Also, we characterize the second fundamental forms of such submanifolds for which the equality cases can hold.

Mohd Danish Siddiqi, Aliya Naaz Siddiqui, Kamran Ahmad

Gravity and Dark Matter in the Framework of Metric-Affine Geometry

Abstract

In this paper, we generalize space-time manifold by considering metric-affine structure on space-time manifold. Metric-affine geometry is defined in terms of two independent objects: the Riemannian metric and the general affine connection. With the help of the metric tensor for contraction of Riemannain curvature of the affine connection, we formed a natural action density for gravity and matter. Using calculus of variations we derive two equations. The first equation retrieves Einstein field equation. The other equation describes matter in space-time. In this framework, the affine connection is related to the concept that is well-known as dark matter, so dark matter can be interpreted as a factor which leads curving and twirling of space-time manifold.

Ghodratallah Fasihi-Ramandi, Vahid Pirhadi

Submersion on Statistical Metallic Structure

Abstract

In the present chapter we define and analyze the properties of metallic structure, i.e., polynomial structures with the polynomial Q(J) = \(J^2\) - aJ - bI, on manifolds using the metallic ratio, which is a generalization of the Golden proportion. We further discuss statistical metallic manifolds and statistical submersions, and we study Riemannian submersion. Also, we give some properties of the metallic Riemannian metric and statistical submersion of the metallic structure.

Mohit Saxena

Tangent Bundles Endowed with Quarter-Symmetric Non-metric -Connection on 3-Dimensional Quasi-Sasakian Manifolds

Abstract

The present paper aims to study the complete lifts of quarter-symmetric non-metric \(\xi \)-connection from a 3-dimensional non-cosymplectic quasi-Sasakian manifold to its tangent bundle and establish specific curvature properties of such connection on the tangent bundle.

Mohammad Nazrul Islam Khan, Ljubica Velimirović

The Darboux Mate and the Higher Order Curvatures of Spherical Legendre Curves

Abstract

We introduce and study a Darboux mate of a given spherical Legendre curve LC for the Euclidean \(S^2\). Also, a triple sequence of curvatures is provided by the higher order derivatives of the last Frenet equation for the frontal of LC. These curvatures are expressed by a recurrence starting with the pair \((-k_1, -k_2, 0)\), where \((k_1, k_2)\) is the classical curvature function of LC. Several examples are discussed, some of them in relationship with the usual theory of regular space curves. The case of Lorentz–Minkowski sphere \(S^2_1\) is sketched only from the point of view of the geodesic curvature.

Mircea Crasmareanu

Conformal -Ricci-Yamabe Solitons in the Framework of Riemannian Manifolds

Abstract

In the present chapter we study Riemannian 3-manifolds \(\mathcal {M}^3\) admitting the conformal \(\eta \)-Ricci-Yamabe solitons (CERYS) and gradient conformal \(\eta \)-Ricci-Yamabe solitons (gradient CERYS). It is proven that if the Riemannian metric of an \(\mathcal {M}^3\) equipped with a semi-symmetric metric \(\zeta \)-connection is a CERYS \((g, F, \sigma , \rho , p, q)\), then the soliton constant is given by \(\sigma =2p-\rho -3q+\frac{1}{2}\left( \psi +\frac{2}{3}\right) +\eta (\pounds _F \zeta )\), provided the scalar curvature of \(\mathcal {M}^3\) is constant. Also, the soliton vector field F of \((g, F, \sigma , \rho , p, q)\) is homothetic if and only if \(\rho +g(\pounds _{F}\zeta , \zeta )=0\). In this sequel, the conditions for the conformal Ricci soliton, conformal Yamabe soliton, conformal Einstein soliton and conformal \(\epsilon \)-Einstein soliton on \(\mathcal {M}^3\) to be expanding, shrinking or steady are established. Next, we prove that an \(\mathcal {M}^3\) endowed with a semi-symmetric metric \(\zeta \)-connection admitting the gradient CERYS \((g, Df, \sigma , \rho , p, q)\) is an Einstein manifold and the gradient of smooth function f is a constant multiple of \(\zeta \). A non-trivial example of an \(\mathcal {M}^3\) equipped with a semi-symmetric metric \(\zeta \)-connection is constructed, and hence verify some of our results.

Sudhakar Kumar Chaubey, Abdul Haseeb

Titel: Geometry of Submanifolds and Applications
herausgegeben von: Bang-Yen Chen
Majid Ali Choudhary
Mohammad Nazrul Islam Khan
Verlag: Springer Nature Singapore
Electronic ISBN: 978-981-9997-50-3
Print ISBN: 978-981-9997-49-7
DOI: https://doi.org/10.1007/978-981-99-9750-3

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Inhaltsverzeichnis

Frontmatter

Recent Developments on Chen–Ricci Inequalities in Differential Geometry

Solitons in -Gravity

A Survey on Lagrangian Submanifolds of Nearly Kaehler Six-Sphere

Pythagorean Submanifolds

On 4-Dimensional Submanifolds of f-Manifolds

Almost Yamabe Solitons on a Total Space of Almost Hermitian Submersions

Generalized Wintgen Inequalities for -Para Sasakian Manifold

Certain Optimal Inequalities for Casorati Curvatures in Quaternion Geometry

Gravity and Dark Matter in the Framework of Metric-Affine Geometry

Submersion on Statistical Metallic Structure

Tangent Bundles Endowed with Quarter-Symmetric Non-metric -Connection on 3-Dimensional Quasi-Sasakian Manifolds

The Darboux Mate and the Higher Order Curvatures of Spherical Legendre Curves

Conformal -Ricci-Yamabe Solitons in the Framework of Riemannian Manifolds

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