Research

6 pages, 249 KiB

Open AccessArticle

On the Extrinsic Principal Directions and Curvatures of Lagrangian Submanifolds

by Marilena Moruz and Leopold Verstraelen

Mathematics 2020, 8(9), 1533; https://doi.org/10.3390/math8091533 - 08 Sep 2020

Cited by 2 | Viewed by 1326

From the basic geometry of submanifolds will be recalled what are the extrinsic principal tangential directions, (first studied by Camille Jordan in the 18seventies), and what are the principal first normal directions, (first studied by Kostadin Trenčevski in the 19nineties), and [...] Read more.

From the basic geometry of submanifolds will be recalled what are the extrinsic principal tangential directions, (first studied by Camille Jordan in the 18seventies), and what are the principal first normal directions, (first studied by Kostadin Trenčevski in the 19nineties), and what are their corresponding Casorati curvatures. For reasons of simplicity of exposition only, hereafter this will merely be done explicitly in the case of arbitrary submanifolds in Euclidean spaces. Then, for the special case of Lagrangian submanifolds in complex Euclidean spaces, the natural relationships between these distinguished tangential and normal directions and their corresponding curvatures will be established. Full article

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

11 pages, 270 KiB

Open AccessArticle

The Minimal Perimeter of a Log-Concave Function

by Niufa Fang and Zengle Zhang

Mathematics 2020, 8(8), 1365; https://doi.org/10.3390/math8081365 - 14 Aug 2020

Viewed by 1686

Abstract

Inspired by the equivalence between isoperimetric inequality and Sobolev inequality, we provide a new connection between geometry and analysis. We define the minimal perimeter of a log-concave function and establish a characteristic theorem of this extremal problem for log-concave functions analogous to convex [...] Read more.

Inspired by the equivalence between isoperimetric inequality and Sobolev inequality, we provide a new connection between geometry and analysis. We define the minimal perimeter of a log-concave function and establish a characteristic theorem of this extremal problem for log-concave functions analogous to convex bodies. Full article

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

8 pages, 235 KiB

Open AccessArticle

Almost Hermitian Identities

by Joana Cirici and Scott O. Wilson

Mathematics 2020, 8(8), 1357; https://doi.org/10.3390/math8081357 - 13 Aug 2020

Cited by 4 | Viewed by 1935

Abstract

We study the local commutation relation between the Lefschetz operator and the exterior differential on an almost complex manifold with a compatible metric. The identity that we obtain generalizes the backbone of the local Kähler identities to the setting of almost Hermitian manifolds, [...] Read more.

We study the local commutation relation between the Lefschetz operator and the exterior differential on an almost complex manifold with a compatible metric. The identity that we obtain generalizes the backbone of the local Kähler identities to the setting of almost Hermitian manifolds, allowing for new global results for such manifolds. Full article

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

19 pages, 796 KiB

Open AccessArticle

Ricci Curvature Inequalities for Skew CR-Warped Product Submanifolds in Complex Space Forms

by Meraj Ali Khan and Ibrahim Aldayel

Mathematics 2020, 8(8), 1317; https://doi.org/10.3390/math8081317 - 07 Aug 2020

Cited by 2 | Viewed by 1495

Abstract

The fundamental goal of this study was to achieve the Ricci curvature inequalities for a skew CR-warped product (SCR W-P) submanifold isometrically immersed in a complex space form (CSF) in the expressions of the squared norm of mean curvature vector and warping functions [...] Read more.

The fundamental goal of this study was to achieve the Ricci curvature inequalities for a skew CR-warped product (SCR W-P) submanifold isometrically immersed in a complex space form (CSF) in the expressions of the squared norm of mean curvature vector and warping functions (W-F). The equality cases were likewise examined. In particular, we also derived Ricci curvature inequalities for CR-warped product (CR W-P) submanifolds. To sustain this study, an example of these submanifolds is provided. Full article

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

11 pages, 252 KiB

Open AccessArticle

Geometric Inequalities of Warped Product Submanifolds and Their Applications

by Nadia Alluhaibi, Fatemah Mofarreh, Akram Ali and Wan Ainun Mior Othman

Mathematics 2020, 8(5), 759; https://doi.org/10.3390/math8050759 - 11 May 2020

Cited by 5 | Viewed by 1804

Abstract

In the present paper, we prove that if Laplacian for the warping function of complete warped product submanifold

M^{m} = B^{p} \times_{h} F^{q}

in a unit sphere

S^{m + k}

satisfies some extrinsic inequalities depending on the dimensions [...] Read more.

In the present paper, we prove that if Laplacian for the warping function of complete warped product submanifold

M^{m} = B^{p} \times_{h} F^{q}

in a unit sphere

S^{m + k}

satisfies some extrinsic inequalities depending on the dimensions of the base

B^{p}

and fiber

F^{q}

such that the base

B^{p}

is minimal, then

M^{m}

must be diffeomorphic to a unit sphere

S^{m}

. Moreover, we give some geometrical classification in terms of Euler–Lagrange equation and Hamiltonian of the warped function. We also discuss some related results. Full article

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

15 pages, 281 KiB

Open AccessArticle

δ(2,2)-Invariant for Lagrangian Submanifolds in Quaternionic Space Forms

by Gabriel Macsim, Adela Mihai and Ion Mihai

Mathematics 2020, 8(4), 480; https://doi.org/10.3390/math8040480 - 01 Apr 2020

Cited by 4 | Viewed by 1657

Abstract

In the geometry of submanifolds, Chen inequalities represent one of the most important tool to find relationships between intrinsic and extrinsic invariants; the aim is to find sharp such inequalities. In this paper we establish an optimal inequality for the Chen invariant [...] Read more.

In the geometry of submanifolds, Chen inequalities represent one of the most important tool to find relationships between intrinsic and extrinsic invariants; the aim is to find sharp such inequalities. In this paper we establish an optimal inequality for the Chen invariant

δ (2, 2)

on Lagrangian submanifolds in quaternionic space forms, regarded as a problem of constrained maxima. Full article

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

9 pages, 243 KiB

Open AccessArticle

A Note on Minimal Hypersurfaces of an Odd Dimensional Sphere

by Sharief Deshmukh and Ibrahim Al-Dayel

Mathematics 2020, 8(2), 294; https://doi.org/10.3390/math8020294 - 21 Feb 2020

Cited by 3 | Viewed by 1839

Abstract

We obtain the Wang-type integral inequalities for compact minimal hypersurfaces in the unit sphere

S^{2 n + 1}

with Sasakian structure and use these inequalities to find two characterizations of minimal Clifford hypersurfaces in the unit sphere [...] Read more.

We obtain the Wang-type integral inequalities for compact minimal hypersurfaces in the unit sphere

S^{2 n + 1}

with Sasakian structure and use these inequalities to find two characterizations of minimal Clifford hypersurfaces in the unit sphere

S^{2 n + 1}

. Full article

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

13 pages, 267 KiB

Open AccessArticle

Inequalities for the Casorati Curvature of Statistical Manifolds in Holomorphic Statistical Manifolds of Constant Holomorphic Curvature

by Simona Decu, Stefan Haesen and Leopold Verstraelen

Mathematics 2020, 8(2), 251; https://doi.org/10.3390/math8020251 - 14 Feb 2020

Cited by 13 | Viewed by 2044

Abstract

In this paper, we prove some inequalities in terms of the normalized

δ

-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of statistical submanifolds in holomorphic statistical manifolds with constant holomorphic sectional curvature. Moreover, we study the equality cases of such [...] Read more.

In this paper, we prove some inequalities in terms of the normalized

δ

-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of statistical submanifolds in holomorphic statistical manifolds with constant holomorphic sectional curvature. Moreover, we study the equality cases of such inequalities. An example on these submanifolds is presented. Full article

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

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 2029

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)

15 pages, 280 KiB

Open AccessArticle

A Closed Form for Slant Submanifolds of Generalized Sasakian Space Forms

by Pablo Alegre, Joaquín Barrera and Alfonso Carriazo

Mathematics 2019, 7(12), 1238; https://doi.org/10.3390/math7121238 - 13 Dec 2019

Cited by 1 | Viewed by 1855

Abstract

The Maslov form is a closed form for a Lagrangian submanifold of

C^{m}

, and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the [...] Read more.

The Maslov form is a closed form for a Lagrangian submanifold of

C^{m}

, and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the scalar curvature, and it happens if and only if the second fundamental form satisfies a certain relation. In a previous paper we presented a natural inequality between the norm of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space forms, characterizing the equality case by certain expression of the second fundamental form. In this paper, first, we present an adapted form for slant submanifolds of a generalized Sasakian space form, similar to the Maslov form, that is always closed. And, in the equality case, we studied under which circumstances the given closed form is also conformal. Full article

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

6 pages, 253 KiB

Open AccessArticle

On the Betti and Tachibana Numbers of Compact Einstein Manifolds

by Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok

Mathematics 2019, 7(12), 1210; https://doi.org/10.3390/math7121210 - 09 Dec 2019

Cited by 3 | Viewed by 1924

Abstract

Throughout the history of the study of Einstein manifolds, researchers have sought relationships between the curvature and topology of such manifolds. In this paper, first, we prove that a compact Einstein manifold

(M, g)

with an Einstein constant [...] Read more.

Throughout the history of the study of Einstein manifolds, researchers have sought relationships between the curvature and topology of such manifolds. In this paper, first, we prove that a compact Einstein manifold

(M, g)

with an Einstein constant

α > 0

is a homological sphere when the minimum of its sectional curvatures

> α / (n + 2)

; in particular,

(M, g)

is a spherical space form when the minimum of its sectional curvatures

> α / n

. Second, we prove two propositions (similar to the above ones) for Tachibana numbers of a compact Einstein manifold with

α < 0

. 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 - 01 Dec 2019

Cited by 12 | Viewed by 2547

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)

19 pages, 327 KiB

Open AccessFeature PaperArticle

Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds

by Aliya Naaz Siddiqui, Bang-Yen Chen and Oguzhan Bahadir

Mathematics 2019, 7(9), 797; https://doi.org/10.3390/math7090797 - 01 Sep 2019

Cited by 21 | Viewed by 2702

Abstract

Warped products play crucial roles in differential geometry, as well as in mathematical physics, especially in general relativity. In this article, first we define and study statistical solitons on Ricci-symmetric statistical warped products

R \times_{f} N_{2}

and [...] Read more.

Warped products play crucial roles in differential geometry, as well as in mathematical physics, especially in general relativity. In this article, first we define and study statistical solitons on Ricci-symmetric statistical warped products

R \times_{f} N_{2}

and

N_{1} \times_{f} R

. Second, we study statistical warped products as submanifolds of statistical manifolds. For statistical warped products statistically immersed in a statistical manifold of constant curvature, we prove Chen’s inequality involving scalar curvature, the squared mean curvature, and the Laplacian of warping function (with respect to the Levi–Civita connection). At the end, we establish a relationship between the scalar curvature and the Casorati curvatures in terms of the Laplacian of the warping function for statistical warped product submanifolds in the same ambient space. Full article

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

14 pages, 263 KiB

Open AccessArticle

Quantum Integral Inequalities of Simpson-Type for Strongly Preinvex Functions

by Yongping Deng, Muhammad Uzair Awan and Shanhe Wu

Mathematics 2019, 7(8), 751; https://doi.org/10.3390/math7080751 - 16 Aug 2019

Cited by 25 | Viewed by 2695

Abstract

In this paper, we establish a new q-integral identity, the result is then used to derive two q-integral inequalities of Simpson-type involving strongly preinvex functions. Some special cases of the obtained results are also considered, it is shown that several new [...] Read more.

In this paper, we establish a new q-integral identity, the result is then used to derive two q-integral inequalities of Simpson-type involving strongly preinvex functions. Some special cases of the obtained results are also considered, it is shown that several new and previously known results can be derived via generalized strongly preinvex functions and quantum integrals. Full article

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

12 pages, 338 KiB

Open AccessArticle

New Refinements of the Erdös–Mordell Inequality and Barrow’s Inequality

by Jian Liu

Mathematics 2019, 7(8), 726; https://doi.org/10.3390/math7080726 - 09 Aug 2019

Cited by 2 | Viewed by 2631

Abstract

In this paper, two new refinements of the Erdös–Mordell inequality and three new refinements of Barrow’s inequality are established. Some related interesting conjectures are put forward. Full article

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

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