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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (28 February 2021) | Viewed by 11028

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Dr. Pablo Sebastián Alegre Rueda

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Guest Editor

Department of Economy, Quantitative Methods and Economy History, Statistics and Operational Research Area, Pablo de Olavide University, Carretera Utrera, Km 1. 41089 Sevilla, Spain
Interests: differential geometry

Special Issue Information

Dear Colleagues,

Since they were introduced in the sixties by S. Sasaki, Sasakian manifolds have been intensively studied, even more since the work developed by C.P. Boyer and K. Galicki, showing the important role these manifolds play in mathematical physics.

Different lines of research are being carried out and seem to be of special interest, starting with new advances in the study of Sasakian–Einstein and 3-Sasakian manifolds and their applications to relativity. In addition, in differential geometry, the curvature of a Riemannian manifold plays a fundamental role, and the study of special expressions and symmetries of certain curvature tensors of Sasakian manifolds is particularly of interest. Moreover, there has been a huge number of studies about some special types of submanifolds (minimal, invariant, anti-invariant, slant, etc.) and several invariants and inequalities related to their Riemannian curvature. Finally, there are some structures close to Sasakian structures (trans-Sasakian, para-Sasakian, etc.) that are increasingly being studied, in both Riemannian and semi-Riemannian geometry.

The purpose of this Special Issue is to provide a collection of papers that reflect new advances in Sasakian spaces, new topics of research, and explore applications in other areas.

Assoc. Prof. Pablo Sebastián Alegre Rueda
Guest Editor

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Keywords

Sasakian manifolds
Sasaki–Einstein manifolds
Submanifolds
Sectional curvature

Published Papers (5 papers)

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Research

17 pages, 302 KiB

Open AccessArticle

On Killing Vector Fields on Riemannian Manifolds

by Sharief Deshmukh and Olga Belova

Mathematics 2021, 9(3), 259; https://doi.org/10.3390/math9030259 - 28 Jan 2021

Cited by 10 | Viewed by 2819

Abstract

We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field

w

on a connected Riemannian manifold

(M, g)

we show that for each non-constant smooth function [...] Read more.

We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field

w

on a connected Riemannian manifold

(M, g)

we show that for each non-constant smooth function

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

there exists a non-zero vector field

w^{f}

associated with f. In particular, we show that for an eigenfunction f of the Laplace operator on an n-dimensional compact Riemannian manifold

(M, g)

with an appropriate lower bound on the integral of the Ricci curvature

S (w^{f}, w^{f})

gives a characterization of the odd-dimensional unit sphere

S^{2 m + 1}

. Also, we show on an n-dimensional compact Riemannian manifold

(M, g)

that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue

n c

and the unit Killing vector field

w

satisfying

{∥\nabla w∥}^{2} \leq (n - 1) c

and Ricci curvature in the direction of the vector field

\nabla f - w

is bounded below by

(n - 1) c

is necessary and sufficient for

(M, g)

to be isometric to the sphere

S^{2 m + 1} (c)

. Finally, we show that the presence of a unit Killing vector field

w

on an n-dimensional Riemannian manifold

(M, g)

with sectional curvatures of plane sections containing

w

equal to 1 forces dimension n to be odd and that the Riemannian manifold

(M, g)

becomes a K-contact manifold. We also show that if in addition

(M, g)

is complete and the Ricci operator satisfies Codazzi-type equation, then

(M, g)

is an Einstein Sasakian manifold. Full article

(This article belongs to the Special Issue Sasakian Space)

9 pages, 254 KiB

Open AccessArticle

Transversal Jacobi Operators in Almost Contact Manifolds

by Jong Taek Cho and Makoto Kimura

Mathematics 2021, 9(1), 31; https://doi.org/10.3390/math9010031 - 24 Dec 2020

Viewed by 1504

Abstract

Along a transversal geodesic

γ

whose tangent belongs to the contact distribution D, we define the transversal Jacobi operator

R_{γ} = R (\cdot, \dot{γ}) \dot{γ}

on an almost contact Riemannian manifold M. Then, using the [...] Read more.

Along a transversal geodesic

γ

whose tangent belongs to the contact distribution D, we define the transversal Jacobi operator

R_{γ} = R (\cdot, \dot{γ}) \dot{γ}

on an almost contact Riemannian manifold M. Then, using the transversal Jacobi operator

R_{γ}

, we give a new characterization of the Sasakian sphere. In the second part, we characterize the complete ruled real hypersurfaces in complex hyperbolic space. Full article

(This article belongs to the Special Issue Sasakian Space)

11 pages, 262 KiB

Open AccessFeature PaperArticle

Metric f-Contact Manifolds Satisfying the (κ, μ)-Nullity Condition

by Alfonso Carriazo, Luis M. Fernández and Eugenia Loiudice

Mathematics 2020, 8(6), 891; https://doi.org/10.3390/math8060891 - 2 Jun 2020

Viewed by 1663

Abstract

We prove that if the f-sectional curvature at any point of a

(2 n + s)

-dimensional metric f-contact manifold satisfying the

(κ, μ)

nullity condition with

n > 1

is independent of the f-section [...] Read more.

We prove that if the f-sectional curvature at any point of a

(2 n + s)

-dimensional metric f-contact manifold satisfying the

(κ, μ)

nullity condition with

n > 1

is independent of the f-section at the point, then it is constant on the manifold. Moreover, we also prove that a non-normal metric f-contact manifold satisfying the

(κ, μ)

nullity condition is of constant f-sectional curvature if and only if

μ = κ + 1

and we give an explicit expression for the curvature tensor field in such a case. Finally, we present some examples. Full article

(This article belongs to the Special Issue Sasakian Space)

11 pages, 775 KiB

Open AccessFeature PaperArticle

Slant Curves in Contact Lorentzian Manifolds with CR Structures

by Ji-Eun Lee

Mathematics 2020, 8(1), 46; https://doi.org/10.3390/math8010046 - 1 Jan 2020

Cited by 6 | Viewed by 1959

Abstract

In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the

\hat{\nabla}

-geodesic is a magnetic curve (for ∇) along slant curves. Finally, we [...] Read more.

In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the

\hat{\nabla}

-geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when

c \leq 0

, there does not exist a non-geodesic slant Frenet curve satisfying the

\hat{\nabla}

-Jacobi equations for the

\hat{\nabla}

-geodesic vector fields in M. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms

M_{1}^{3} (\hat{H})

for

\hat{H} = 2 c > 0

. Full article

(This article belongs to the Special Issue Sasakian Space)

11 pages, 775 KiB

Open AccessArticle

On Jacobi-Type Vector Fields on Riemannian Manifolds

by Bang-Yen Chen, Sharief Deshmukh and Amira A. Ishan

Mathematics 2019, 7(12), 1139; https://doi.org/10.3390/math7121139 - 21 Nov 2019

Cited by 11 | Viewed by 2597

Abstract

In this article, we study Jacobi-type vector fields on Riemannian manifolds. A Killing vector field is a Jacobi-type vector field while the converse is not true, leading to a natural question of finding conditions under which a Jacobi-type vector field is Killing. In this article, we first prove that every Jacobi-type vector field on a compact Riemannian manifold is Killing. Then, we find several necessary and sufficient conditions for a Jacobi-type vector field to be a Killing vector field on non-compact Riemannian manifolds. Further, we derive some characterizations of Euclidean spaces in terms of Jacobi-type vector fields. Finally, we provide examples of Jacobi-type vector fields on non-compact Riemannian manifolds, which are non-Killing. Full article

(This article belongs to the Special Issue Sasakian Space)

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