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7 pages, 243 KiB

Open AccessArticle

On Characterizing a Three-Dimensional Sphere

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

Mathematics 2021, 9(24), 3311; https://doi.org/10.3390/math9243311 - 19 Dec 2021

Cited by 1 | Viewed by 2149

In this paper, we find a characterization of the 3-sphere using 3-dimensional compact and simply connected trans-Sasakian manifolds of type

(α, β)

. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

9 pages, 262 KiB

Open AccessArticle

Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold

by Amira Ishan, Sharief Deshmukh and Gabriel-Eduard Vîlcu

Mathematics 2021, 9(8), 863; https://doi.org/10.3390/math9080863 - 14 Apr 2021

Cited by 4 | Viewed by 2444

Abstract

We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces. We find two new characterizations of the m-dimensional sphere

S^{m} (c)

of constant curvature c. The first characterization uses the well [...] Read more.

We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces. We find two new characterizations of the m-dimensional sphere

S^{m} (c)

of constant curvature c. The first characterization uses the well known de-Rham Laplace operator, while the second uses a nontrivial solution of the famous Fischer–Marsden differential equation. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: 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 - 4 Feb 2021

Cited by 2 | Viewed by 2469

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)

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