MDPI - Publisher of Open Access Journals

20 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

Cited by 2 | Viewed by 1078

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)

15 pages, 292 KiB

Open AccessArticle

Optimal Inequalities for Submanifolds in Trans-Sasakian Manifolds Endowed with a Semi-Symmetric Metric Connection

by Ion Mihai and Mohammed Mohammed

Symmetry 2023, 15(4), 877; https://doi.org/10.3390/sym15040877 - 6 Apr 2023

Cited by 7 | Viewed by 1407

Abstract

In this paper, we improve the Chen first inequality for special contact slant submanifolds and Legendrian submanifolds, respectively, in (

α, β

) trans-Sasakian generalized Sasakian space forms endowed with a semi-symmetric metric connection. Full article

(This article belongs to the Special Issue Topological Graph Theory and Discrete Geometry II)

18 pages, 330 KiB

Open AccessArticle

Some Pinching Results for Bi-Slant Submanifolds in S-Space Forms

by Mohd Aquib, Meraj Ali Khan, Adela Mihai and Ion Mihai

Mathematics 2022, 10(9), 1538; https://doi.org/10.3390/math10091538 - 3 May 2022

Viewed by 1744

Abstract

The objective of the present article is to prove two geometric inequalities for submanifolds in S-space forms. First, we establish inequalities for the generalized normalized

δ

-Casorati curvatures for bi-slant submanifolds in S-space forms and then we derive the generalized Wintgen [...] Read more.

The objective of the present article is to prove two geometric inequalities for submanifolds in S-space forms. First, we establish inequalities for the generalized normalized

δ

-Casorati curvatures for bi-slant submanifolds in S-space forms and then we derive the generalized Wintgen inequality for Legendrian and bi-slant submanifolds in the same ambient space. We also discuss the equality cases of the inequalities. Further, we provide some immediate geometric applications of the results. Finally, we construct some examples of slant and Legendrian submanifolds, respectively. Full article

(This article belongs to the Special Issue Complex and Contact Manifolds II)

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 1443

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)

41 pages, 526 KiB

Open AccessArticle

Contact Dynamics: Legendrian and Lagrangian Submanifolds

by Oğul Esen, Manuel Lainz Valcázar, Manuel de León and Juan Carlos Marrero

Mathematics 2021, 9(21), 2704; https://doi.org/10.3390/math9212704 - 25 Oct 2021

Cited by 13 | Viewed by 3121

Abstract

We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a [...] Read more.

We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew’s triple is constructed for evolution contact dynamics. Full article

(This article belongs to the Special Issue New Trends in Hamilton-Jacobi Theory: Conservative and Dissipative Dynamics)

19 pages, 313 KiB

Open AccessArticle

Differential Invariants of Measurements, and Their Relation to Central Moments

by Eivind Schneider

Entropy 2020, 22(10), 1118; https://doi.org/10.3390/e22101118 - 3 Oct 2020

Cited by 5 | Viewed by 1705

Abstract

Due to the principle of minimal information gain, the measurement of points in an affine space V determines a Legendrian submanifold of

V \times V^{*} \times R

. Such Legendrian submanifolds are equipped with additional geometric structures that come from the central [...] Read more.

Due to the principle of minimal information gain, the measurement of points in an affine space V determines a Legendrian submanifold of

V \times V^{*} \times R

. Such Legendrian submanifolds are equipped with additional geometric structures that come from the central moments of the underlying probability distributions and are invariant under the action of the group of affine transformations on V. We investigate the action of this group of affine transformations on Legendrian submanifolds of

V \times V^{*} \times R

by giving a detailed overview of the structure of the algebra of scalar differential invariants, and we show how the scalar differential invariants can be constructed from the central moments. In the end, we view the results in the context of equilibrium thermodynamics of gases, and notice that the heat capacity is one of the differential invariants. Full article

(This article belongs to the Special Issue Thermodynamics, Geometry and Control Theory)

14 pages, 784 KiB

Open AccessArticle

Optimal Thermodynamic Processes For Gases

by Alexei Kushner, Valentin Lychagin and Mikhail Roop

Entropy 2020, 22(4), 448; https://doi.org/10.3390/e22040448 - 15 Apr 2020

Cited by 15 | Viewed by 3088

Abstract

In this paper, we consider an optimal control problem in the equilibrium thermodynamics of gases. The thermodynamic state of the gas is given by a Legendrian submanifold in a contact thermodynamic space. Using Pontryagin’s maximum principle, we find a thermodynamic process in this [...] Read more.

In this paper, we consider an optimal control problem in the equilibrium thermodynamics of gases. The thermodynamic state of the gas is given by a Legendrian submanifold in a contact thermodynamic space. Using Pontryagin’s maximum principle, we find a thermodynamic process in this submanifold such that the gas maximizes the work functional. For ideal gases, this problem is shown to be integrable in Liouville’s sense and its solution is given by means of action-angle variables. For real gases considered to be a perturbation of ideal ones, the integrals are given asymptotically. Full article

(This article belongs to the Special Issue Thermodynamics, Geometry and Control Theory)

10 pages, 264 KiB

Open AccessArticle

On Differential Equations Characterizing Legendrian Submanifolds of Sasakian Space Forms

by Rifaqat Ali, Fatemah Mofarreh, Nadia Alluhaibi, Akram Ali and Iqbal Ahmad

Mathematics 2020, 8(2), 150; https://doi.org/10.3390/math8020150 - 21 Jan 2020

Cited by 14 | Viewed by 2579

Abstract

In this paper, we give an estimate of the first eigenvalue of the Laplace operator on minimally immersed Legendrian submanifold

N^{n}

in Sasakian space forms

{\tilde{N}}^{2 n + 1} (ϵ)

. We prove that a minimal Legendrian submanifolds [...] Read more.

In this paper, we give an estimate of the first eigenvalue of the Laplace operator on minimally immersed Legendrian submanifold

N^{n}

in Sasakian space forms

{\tilde{N}}^{2 n + 1} (ϵ)

. We prove that a minimal Legendrian submanifolds in a Sasakian space form is isometric to a standard sphere

S^{n}

if the Ricci curvature satisfies an extrinsic condition which includes a gradient of a function, the constant holomorphic sectional curvature of the ambient space and a dimension of

N^{n}

. We also obtain a Simons-type inequality for the same ambient space forms

{\tilde{N}}^{2 n + 1} (ϵ)

. Full article

(This article belongs to the Special Issue Inequalities in Geometry and Applications)

20 pages, 830 KiB

Open AccessArticle

The First Fundamental Equation and Generalized Wintgen-Type Inequalities for Submanifolds in Generalized Space Forms

by Mohd. Aquib, Michel Nguiffo Boyom, Mohammad Hasan Shahid and Gabriel-Eduard Vîlcu

Mathematics 2019, 7(12), 1151; https://doi.org/10.3390/math7121151 - 1 Dec 2019

Cited by 13 | Viewed by 3139

Abstract

In this work, we first derive a generalized Wintgen type inequality for a Lagrangian submanifold in a generalized complex space form. Further, we extend this inequality to the case of bi-slant submanifolds in generalized complex and generalized Sasakian space forms and derive some [...] Read more.

In this work, we first derive a generalized Wintgen type inequality for a Lagrangian submanifold in a generalized complex space form. Further, we extend this inequality to the case of bi-slant submanifolds in generalized complex and generalized Sasakian space forms and derive some applications in various slant cases. Finally, we obtain obstructions to the existence of non-flat generalized complex space forms and non-flat generalized Sasakian space forms in terms of dimension of the vector space of solutions to the first fundamental equation on such spaces. Full article

(This article belongs to the Special Issue Inequalities in Geometry and Applications)

10 pages, 248 KiB

Open AccessArticle

Inequalities on Sasakian Statistical Manifolds in Terms of Casorati Curvatures

by Chul Woo Lee and Jae Won Lee

Mathematics 2018, 6(11), 259; https://doi.org/10.3390/math6110259 - 17 Nov 2018

Cited by 5 | Viewed by 2726

Abstract

A statistical structure is considered as a generalization of a pair of a Riemannian metric and its Levi-Civita connection. With a pair of conjugate connections ∇ and

\nabla^{*}

in the Sasakian statistical structure, we provide the normalized scalar curvature which is bounded [...] Read more.

A statistical structure is considered as a generalization of a pair of a Riemannian metric and its Levi-Civita connection. With a pair of conjugate connections ∇ and

\nabla^{*}

in the Sasakian statistical structure, we provide the normalized scalar curvature which is bounded above from Casorati curvatures on C-totally real (Legendrian and slant) submanifolds of a Sasakian statistical manifold of constant

φ

-sectional curvature. In addition, we give examples to show that the total space is a sphere. Full article

(This article belongs to the Special Issue Differential Geometry)

15 pages, 284 KiB

Open AccessArticle

Open Gromov-Witten Invariants from the Augmentation Polynomial

by Matthew Mahowald

Symmetry 2017, 9(10), 232; https://doi.org/10.3390/sym9100232 - 17 Oct 2017

Viewed by 3638

Abstract

A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of

X = O_{P^{1}} (- 1, - 1)

to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, [...] Read more.

A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of

X = O_{P^{1}} (- 1, - 1)

to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, one boundary component open Gromov-Witten invariants for Lagrangian submanifolds

L_{K} \subset X

obtained from the conormal bundles of knots

K \subset S^{3}

. This computation is then performed for two non-toric examples (the figure-eight and three-twist knots). For

(r, s)

torus knots, the open Gromov-Witten invariants can also be computed using Atiyah-Bott localization. Using this result for the unknot and the

(3, 2)

torus knot, we show that the augmentation polynomial can be derived from these open Gromov-Witten invariants. Full article

(This article belongs to the Special Issue Knot Theory and Its Applications)

► Show Figures

Figure 1

Search Results (11)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (11)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI