Research

15 pages, 308 KiB

Open AccessArticle

On the Canonical Foliation of an Indefinite Locally Conformal Kähler Manifold with a Parallel Lee Form

by Elisabetta Barletta, Sorin Dragomir and Francesco Esposito

Mathematics 2021, 9(4), 333; https://doi.org/10.3390/math9040333 - 07 Feb 2021

Viewed by 1100

We study the semi-Riemannian geometry of the foliation

F

of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation

ω = 0

, provided that

\nabla ω = 0

and

c = ∥ ω ∥ \neq 0

( [...] Read more.

We study the semi-Riemannian geometry of the foliation

F

of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation

ω = 0

, provided that

\nabla ω = 0

and

c = ∥ ω ∥ \neq 0

(

ω

is the Lee form of M). If M is conformally flat then every leaf of

F

is shown to be a totally geodesic semi-Riemannian hypersurface in M, and a semi-Riemannian space form of sectional curvature

c / 4

, carrying an indefinite c-Sasakian structure. As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem any geodesically complete, conformally flat, indefinite Vaisman manifold of index

2 s

,

0 < s < n

, is locally biholomorphically homothetic to an indefinite complex Hopf manifold

C H_{s}^{n} (λ)

,

0 < λ < 1

, equipped with the indefinite Boothby metric

g_{s, n}

. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

13 pages, 291 KiB

Open AccessArticle

A Note on Killing Calculus on Riemannian Manifolds

by Sharief Deshmukh, Amira Ishan, Suha B. Al-Shaikh and Cihan Özgür

Mathematics 2021, 9(4), 307; https://doi.org/10.3390/math9040307 - 04 Feb 2021

Cited by 2 | Viewed by 1850

Abstract

In this article, it has been observed that a unit Killing vector field

ξ

on an n-dimensional Riemannian manifold

(M, g)

, influences its algebra of smooth functions

C^{\infty} (M)

. For instance, if h is [...] Read more.

In this article, it has been observed that a unit Killing vector field

ξ

on an n-dimensional Riemannian manifold

(M, g)

, influences its algebra of smooth functions

C^{\infty} (M)

. For instance, if h is an eigenfunction of the Laplace operator

Δ

with eigenvalue

λ

, then

ξ (h)

is also eigenfunction with same eigenvalue. Additionally, it has been observed that the Hessian

H^{h} (ξ, ξ)

of a smooth function

h \in C^{\infty} (M)

defines a self adjoint operator

⊡_{ξ}

and has properties similar to most of properties of the Laplace operator on a compact Riemannian manifold

(M, g)

. We study several properties of functions associated to the unit Killing vector field

ξ

. Finally, we find characterizations of the odd dimensional sphere using properties of the operator

⊡_{ξ}

and the nontrivial solution of Fischer–Marsden differential equation, respectively. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

11 pages, 284 KiB

Open AccessArticle

On a Metric Affine Manifold with Several Orthogonal Complementary Distributions

by Vladimir Rovenski and Sergey E. Stepanov

Mathematics 2021, 9(3), 229; https://doi.org/10.3390/math9030229 - 25 Jan 2021

Cited by 1 | Viewed by 1425

Abstract

A Riemannian manifold endowed with

k > 2

orthogonal complementary distributions (called here an almost multi-product structure) appears in such topics as multiply twisted or warped products and the webs or nets composed of orthogonal foliations. In this article, we define the mixed [...] Read more.

A Riemannian manifold endowed with

k > 2

orthogonal complementary distributions (called here an almost multi-product structure) appears in such topics as multiply twisted or warped products and the webs or nets composed of orthogonal foliations. In this article, we define the mixed scalar curvature of an almost multi-product structure endowed with a linear connection, and represent this kind of curvature using fundamental tensors of distributions and the divergence of a geometrically interesting vector field. Using this formula, we prove decomposition and non-existence theorems and integral formulas that generalize results (for

k = 2

) on almost product manifolds with the Levi-Civita connection. Some of our results are illustrated by examples with statistical and semi-symmetric connections. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

11 pages, 242 KiB

Open AccessArticle

Representation of Maximal Surfaces in a 3-Dimensional Lightlike Cone

by Jinhua Qian, Xueshan Fu and Huili Liu

Mathematics 2020, 8(11), 1921; https://doi.org/10.3390/math8111921 - 02 Nov 2020

Viewed by 1240

Abstract

In this paper, the representation formula of maximal surfaces in a 3-dimensional lightlike cone

Q^{3}

is obtained by making use of the differential equation theory and complex function theory. Some particular maximal surfaces under a special induced metric are presented explicitly via [...] Read more.

In this paper, the representation formula of maximal surfaces in a 3-dimensional lightlike cone

Q^{3}

is obtained by making use of the differential equation theory and complex function theory. Some particular maximal surfaces under a special induced metric are presented explicitly via the representation formula. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

17 pages, 296 KiB

Open AccessArticle

Analytic Extension of Riemannian Analytic Manifolds and Local Isometries

by Vladimir A. Popov

Mathematics 2020, 8(11), 1855; https://doi.org/10.3390/math8111855 - 22 Oct 2020

Cited by 2 | Viewed by 1299

Abstract

This article deals with a locally given Riemannian analytic manifold. One of the main tasks is to define its regular analytic extension in order to generalize the notion of completeness. Such extension is studied for metrics whose Lie algebra of all Killing vector [...] Read more.

This article deals with a locally given Riemannian analytic manifold. One of the main tasks is to define its regular analytic extension in order to generalize the notion of completeness. Such extension is studied for metrics whose Lie algebra of all Killing vector fields has no center. The generalization of completeness for an arbitrary metric is given, too. Another task is to analyze the possibility of extending local isometry to isometry of some manifold. It can be done for metrics whose Lie algebra of all Killing vector fields has no center. For such metrics there exists a manifold on which any Killing vector field generates one parameter group of isometries. We prove the following almost necessary condition under which Lie algebra of all Killing vector fields generates a group of isometries on some manifold. Let

g

be Lie algebra of all Killing vector fields on Riemannian analytic manifold,

h \subset g

is its stationary subalgebra,

z \subset g

is its center and [

g, g]

is commutant.

G

is Lie group generated by

g

and is subgroup generated by

h \subset g

. If

h \cap (z + [g; g]) = h \cap [g; g]

, then

H

is closed in

G

. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

16 pages, 322 KiB

Open AccessArticle

Geometric Inequalities of Bi-Warped Product Submanifolds of Nearly Kenmotsu Manifolds and Their Applications

by Akram Ali and Fatemah Mofarreh

Mathematics 2020, 8(10), 1805; https://doi.org/10.3390/math8101805 - 16 Oct 2020

Cited by 4 | Viewed by 1480

Abstract

The present paper aims to construct an inequality for bi-warped product submanifolds in a special class of almost metric manifolds, namely nearly Kenmotsu manifolds. As geometric applications, some exceptional cases that generalized several other inequalities are discussed. We also deliberate some applications in [...] Read more.

The present paper aims to construct an inequality for bi-warped product submanifolds in a special class of almost metric manifolds, namely nearly Kenmotsu manifolds. As geometric applications, some exceptional cases that generalized several other inequalities are discussed. We also deliberate some applications in the context of mathematical physics and derive a new relation between the Dirichlet energy and the second fundamental form. Finally, we present a constructive remark at the end of this paper which shows the motive of the study. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

12 pages, 279 KiB

Open AccessArticle

Necessary and Sufficient Second-Order Optimality Conditions on Hadamard Manifolds

by Gabriel Ruiz-Garzón, Jaime Ruiz-Zapatero, Rafaela Osuna-Gómez and Antonio Rufián-Lizana

Mathematics 2020, 8(7), 1152; https://doi.org/10.3390/math8071152 - 14 Jul 2020

Cited by 3 | Viewed by 1690

Abstract

This work is intended to lead a study of necessary and sufficient optimality conditions for scalar optimization problems on Hadamard manifolds. In the context of this geometry, we obtain and present new function types characterized by the property of having all their second-order [...] Read more.

This work is intended to lead a study of necessary and sufficient optimality conditions for scalar optimization problems on Hadamard manifolds. In the context of this geometry, we obtain and present new function types characterized by the property of having all their second-order stationary points be global minimums. In order to do so, we extend the concept convexity in Euclidean space to a more general notion of invexity on Hadamard manifolds. This is done employing notions of second-order directional derivatives, second-order pseudoinvexity functions, and the second-order Karush–Kuhn–Tucker-pseudoinvexity problem. Thus, we prove that every second-order stationary point is a global minimum if and only if the problem is either second-order pseudoinvex or second-order KKT-pseudoinvex depending on whether the problem regards unconstrained or constrained scalar optimization, respectively. This result has not been presented in the literature before. Finally, examples of these new characterizations are provided in the context of “Higgs Boson like” potentials, among others. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

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33 pages, 410 KiB

Open AccessArticle

Pointwise Slant and Pointwise Semi-Slant Submanifolds in Almost Contact Metric Manifolds

by Kwang Soon Park

Mathematics 2020, 8(6), 985; https://doi.org/10.3390/math8060985 - 16 Jun 2020

Cited by 13 | Viewed by 2496

Abstract

In almost contact metric manifolds, we consider two kinds of submanifolds: pointwise slant, pointwise semi-slant. On these submanifolds of cosymplectic, Sasakian and Kenmotsu manifolds, we obtain characterizations and study their topological properties and distributions. We also give their examples. In particular, we obtain [...] Read more.

In almost contact metric manifolds, we consider two kinds of submanifolds: pointwise slant, pointwise semi-slant. On these submanifolds of cosymplectic, Sasakian and Kenmotsu manifolds, we obtain characterizations and study their topological properties and distributions. We also give their examples. In particular, we obtain some inequalities consisting of a second fundamental form, a warping function and a semi-slant function. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

9 pages, 246 KiB

Open AccessArticle

Generalized Sasakian Space Forms Which Are Realized as Real Hypersurfaces in Complex Space Forms

by Alfonso Carriazo, Jong Taek Cho and Verónica Martín-Molina

Mathematics 2020, 8(6), 873; https://doi.org/10.3390/math8060873 - 29 May 2020

Cited by 4 | Viewed by 1645

Abstract

We prove a classification theorem of the generalized Sasakian space forms which are realized as real hypersurfaces in complex space forms. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

8 pages, 749 KiB

Open AccessArticle

Singular Special Curves in 3-Space Forms

by Jie Huang and Donghe Pei

Mathematics 2020, 8(5), 846; https://doi.org/10.3390/math8050846 - 23 May 2020

Cited by 7 | Viewed by 1936

Abstract

We study the singular Bertrand curves and Mannheim curves in the 3-dimensional space forms. We introduce the geometrical properties of such special curves. Moreover, we get the relationships between singularities of original curves and torsions of another mate curves. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

12 pages, 285 KiB

Open AccessArticle

A Characterization of Ruled Real Hypersurfaces in Non-Flat Complex Space Forms

by George Kaimakamis, Konstantina Panagiotidou and Juan de Dios Pérez

Mathematics 2020, 8(4), 642; https://doi.org/10.3390/math8040642 - 21 Apr 2020

Viewed by 1743

Abstract

The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator [...] Read more.

The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator

F_{X}^{(k)}

is defined and is related to both connections. If X belongs to the maximal holomorphic distribution

D

on M, the corresponding operator does not depend on k and is denoted by

F_{X}

and called Cho operator. In this paper, real hypersurfaces in non-flat space forms such that

F_{X} S = S F_{X}

, where S denotes the Ricci tensor of M and a further condition is satisfied, are classified. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

11 pages, 736 KiB

Open AccessArticle

A New Angular Measurement in Minkowski 3-Space

by Jinhua Qian, Xueqian Tian, Jie Liu and Young Ho Kim

Mathematics 2020, 8(1), 56; https://doi.org/10.3390/math8010056 - 02 Jan 2020

Cited by 1 | Viewed by 1771

Abstract

In Lorentz–Minkowski space, the angles between any two non-null vectors have been defined in the sense of the angles in Euclidean space. In this work, the angles relating to lightlike vectors are characterized by the Frenet frame of a pseudo null curve and [...] Read more.

In Lorentz–Minkowski space, the angles between any two non-null vectors have been defined in the sense of the angles in Euclidean space. In this work, the angles relating to lightlike vectors are characterized by the Frenet frame of a pseudo null curve and the angles between any two non-null vectors in Minkowski 3-space. Meanwhile, the explicit measuring methods are demonstrated through several examples. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

20 pages, 561 KiB

Open AccessFeature PaperArticle

The Dirichlet Problem of the Constant Mean Curvature in Equation in Lorentz-Minkowski Space and in Euclidean Space

by Rafael López

Mathematics 2019, 7(12), 1211; https://doi.org/10.3390/math7121211 - 09 Dec 2019

Cited by 2 | Viewed by 2304

Abstract

We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the [...] Read more.

We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the spacelike condition in the Lorentz-Minkowski space allows dropping the hypothesis on the mean convexity, which is required in the Euclidean case. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

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14 pages, 808 KiB

Open AccessArticle

Characterizations of Positive Operator-Monotone Functions and Monotone Riemannian Metrics via Borel Measures

by Pattrawut Chansangiam and Sorin V. Sabau

Mathematics 2019, 7(12), 1162; https://doi.org/10.3390/math7121162 - 02 Dec 2019

Viewed by 1425

Abstract

We show that there is a one-to-one correspondence between positive operator-monotone functions on the positive reals, monotone Riemannian metrics, and finite positive Borel measures on the unit interval. This correspondence appears as an integral representation of weighted harmonic means with respect to that [...] Read more.

We show that there is a one-to-one correspondence between positive operator-monotone functions on the positive reals, monotone Riemannian metrics, and finite positive Borel measures on the unit interval. This correspondence appears as an integral representation of weighted harmonic means with respect to that measure on the unit interval. We also investigate the normalized/symmetric conditions for operator-monotone functions. These conditions turn out to characterize monotone metrics and Morozowa–Chentsov functions as well. Concrete integral representations of such functions related to well-known monotone metrics are also provided. Moreover, we use this integral representation to decompose positive operator-monotone functions. Such decomposition gives rise to a decomposition of the associated monotone metric. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

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