Mathematics

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27 pages, 333 KiB

Open AccessArticle

The Geometry of a Randers Rotational Surface with an Arbitrary Direction Wind

by Rattanasak Hama and Sorin V. Sabau

Mathematics 2020, 8(11), 2047; https://doi.org/10.3390/math8112047 - 17 Nov 2020

Cited by 4 | Viewed by 1565

In the present paper, we study the global behaviour of geodesics of a Randers metric, defined on Finsler surfaces of revolution, obtained as the solution of the Zermelo’s navigation problem. Our wind is not necessarily a Killing field. We apply our findings to [...] Read more.

In the present paper, we study the global behaviour of geodesics of a Randers metric, defined on Finsler surfaces of revolution, obtained as the solution of the Zermelo’s navigation problem. Our wind is not necessarily a Killing field. We apply our findings to the case of the topological cylinder

R \times S^{1}

and describe in detail the geodesics behaviour, the conjugate and cut loci. Full article

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

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11 pages, 279 KiB

Open AccessArticle

A Geometric Obstruction for CR-Slant Warped Products in a Nearly Cosymplectic Manifold

by Siraj Uddin and M. Z. Ullah

Mathematics 2020, 8(9), 1622; https://doi.org/10.3390/math8091622 - 19 Sep 2020

Cited by 1 | Viewed by 1657

Abstract

In the early 20th century, B.-Y. Chen introduced the concept of CR-warped products and obtained several fundamental results, such as inequality for the length of second fundamental form. In this paper, we obtain B.-Y. Chen’s inequality for CR-slant warped products in nearly cosymplectic [...] Read more.

In the early 20th century, B.-Y. Chen introduced the concept of CR-warped products and obtained several fundamental results, such as inequality for the length of second fundamental form. In this paper, we obtain B.-Y. Chen’s inequality for CR-slant warped products in nearly cosymplectic manifolds, which are the more general classes of manifolds. The equality case of this inequality is also investigated. Furthermore, the inequality is discussed for some important subclasses of CR-slant warped products. Full article

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

13 pages, 256 KiB

Open AccessArticle

Contact Metric Spaces and pseudo-Hermitian Symmetry

by Jong Taek Cho

Mathematics 2020, 8(9), 1583; https://doi.org/10.3390/math8091583 - 14 Sep 2020

Viewed by 1624

Abstract

We prove that a contact strongly pseudo-convex CR (Cauchy–Riemann) manifold

M^{2 n + 1}

,

n \geq 2

, is locally pseudo-Hermitian symmetric and satisfies

\nabla_{ξ} h = μ h ϕ

,

μ \in R

, if and only if M [...] Read more.

We prove that a contact strongly pseudo-convex CR (Cauchy–Riemann) manifold

M^{2 n + 1}

,

n \geq 2

, is locally pseudo-Hermitian symmetric and satisfies

\nabla_{ξ} h = μ h ϕ

,

μ \in R

, if and only if M is either a Sasakian locally

ϕ

-symmetric space or a non-Sasakian

(k, μ)

-space. When

n = 1

, we prove a classification theorem of contact strongly pseudo-convex CR manifolds with pseudo-Hermitian symmetry. Full article

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

17 pages, 296 KiB

Open AccessArticle

Hypersurfaces of a Sasakian Manifold

by Haila Alodan, Sharief Deshmukh, Nasser Bin Turki and Gabriel-Eduard Vîlcu

Mathematics 2020, 8(6), 877; https://doi.org/10.3390/math8060877 - 01 Jun 2020

Cited by 9 | Viewed by 1811

Abstract

We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field

ξ

of the Sasakian manifold induces a vector field

ξ^{T}

on the hypersurface, namely the tangential component of

ξ

to hypersurface, and it also [...] Read more.

We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field

ξ

of the Sasakian manifold induces a vector field

ξ^{T}

on the hypersurface, namely the tangential component of

ξ

to hypersurface, and it also gives a smooth function

ρ

on the hypersurface, which is the projection of the Reeb vector field on the unit normal. First, we find volume estimates for a compact orientable hypersurface and then we use them to find an upper bound of the first nonzero eigenvalue of the Laplace operator on the hypersurface, showing that if the equality holds then the hypersurface is isometric to a certain sphere. Also, we use a bound on the energy of the vector field

\nabla ρ

on a compact orientable hypersurface in a Sasakian manifold in order to find another geometric condition (in terms of mean curvature and integral curves of

ξ^{T}

) under which the hypersurface is isometric to a sphere. Finally, we study compact orientable hypersurfaces with constant mean curvature in a Sasakian manifold and find a sharp upper bound on the first nonzero eigenvalue of the Laplace operator on the hypersurface. In particular, we show that this upper bound is attained if and only if the hypersurface is isometric to a sphere, provided that the Ricci curvature of the hypersurface along

\nabla ρ

has a certain lower bound. Full article

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

19 pages, 276 KiB

Open AccessArticle

Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds

by Hülya Aytimur, Mayuko Kon, Adela Mihai, Cihan Özgür and Kazuhiko Takano

Mathematics 2019, 7(12), 1202; https://doi.org/10.3390/math7121202 - 08 Dec 2019

Cited by 18 | Viewed by 2353

Abstract

We consider Kähler-like statistical manifolds, whose curvature tensor field satisfies a natural condition. For their statistical submanifolds, we prove a Chen first inequality and a Chen inequality for the invariant

δ (2, 2)

. [...] Read more.

We consider Kähler-like statistical manifolds, whose curvature tensor field satisfies a natural condition. For their statistical submanifolds, we prove a Chen first inequality and a Chen inequality for the invariant

δ (2, 2)

. Full article

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

11 pages, 259 KiB

Open AccessArticle

Submanifolds in Normal Complex Contact Manifolds

by Adela Mihai and Ion Mihai

Mathematics 2019, 7(12), 1195; https://doi.org/10.3390/math7121195 - 05 Dec 2019

Viewed by 1756

Abstract

In the present article we initiate the study of submanifolds in normal complex contact metric manifolds. We define invariant and anti-invariant (

C C

-totally real) submanifolds in such manifolds and start the study of their basic properties. Also, we establish the Chen [...] Read more.

In the present article we initiate the study of submanifolds in normal complex contact metric manifolds. We define invariant and anti-invariant (

C C

-totally real) submanifolds in such manifolds and start the study of their basic properties. Also, we establish the Chen first inequality and Chen inequality for the invariant

δ (2, 2)

for

C C

-totally real submanifolds in a normal complex contact space form and characterize the equality cases. We also prove the minimality of

C C

-totally real submanifolds of maximum dimension satisfying the equalities. Full article

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

7 pages, 262 KiB

Open AccessArticle

Holomorphic Approximation on Certain Weakly Pseudoconvex Domains in Cⁿ

by Shaban Khidr

Mathematics 2019, 7(11), 1035; https://doi.org/10.3390/math7111035 - 03 Nov 2019

Cited by 1 | Viewed by 1521

Abstract

The purpose of this paper is to study the Mergelyan approximation property in

L^{p}

and

C^{k}

-scales on certain weakly pseudoconvex domains of finite/infinite type in

C^{n}

. At the heart of our results lies the solvability of the additive [...] Read more.

The purpose of this paper is to study the Mergelyan approximation property in

L^{p}

and

C^{k}

-scales on certain weakly pseudoconvex domains of finite/infinite type in

C^{n}

. At the heart of our results lies the solvability of the additive Cousin problem with bounds as well as estimates of the

\bar{\partial}

-equation in the corresponding topologies. Full article

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

10 pages, 247 KiB

Open AccessFeature PaperArticle

On the Sign of the Curvature of a Contact Metric Manifold

by David E. Blair

Mathematics 2019, 7(10), 892; https://doi.org/10.3390/math7100892 - 24 Sep 2019

Cited by 1 | Viewed by 1944

Abstract

In this expository article, we discuss the author’s conjecture that an associated metric for a given contact form on a contact manifold of dimension ≥5 must have some positive curvature. In dimension 3, the standard contact structure on the 3-torus admits a flat [...] Read more.

In this expository article, we discuss the author’s conjecture that an associated metric for a given contact form on a contact manifold of dimension ≥5 must have some positive curvature. In dimension 3, the standard contact structure on the 3-torus admits a flat associated metric; we also discuss a local example, due to Krouglov, where there exists a neighborhood of negative curvature on a particular 3-dimensional contact metric manifold. In the last section, we review some results on contact metric manifolds with negative sectional curvature for sections containing the Reeb vector field. Full article

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

7 pages, 258 KiB

Open AccessFeature PaperArticle

Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

by Bang-Yen Chen

Mathematics 2019, 7(8), 710; https://doi.org/10.3390/math7080710 - 06 Aug 2019

Cited by 8 | Viewed by 2731

Abstract

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in [...] Read more.

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in

E^{3}

. Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in

E^{4}

; and Dimitric proved that biharmonic hypersurfaces in

E^{m}

with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for

δ (2)

-ideal and

δ (3)

-ideal hypersurfaces in

E^{m}

. Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in

E^{m}

with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors. Full article

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

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