On Killing Vector Fields on Riemannian Manifolds

Sharief Deshmukh; Olga Belova

doi:10.3390/math9030259

On Killing Vector Fields on Riemannian Manifolds

Mathematics ◽

10.3390/math9030259 ◽

2021 ◽

Vol 9 (3) ◽

pp. 259

Author(s):

Sharief Deshmukh ◽

Olga Belova

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Laplace Operator ◽

Ricci Curvature ◽

Riemannian Manifolds ◽

Smooth Function ◽

Compact Riemannian Manifold ◽

Killing Vector ◽

Killing Vector Field ◽

The Laplace Operator

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∈C∞(M) there exists a non-zero vector field wf 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(wf,wf) gives a characterization of the odd-dimensional unit sphere S2m+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 nc and the unit Killing vector field w satisfying ∇w2≤(n−1)c and Ricci curvature in the direction of the vector field ∇f−w is bounded below by n−1c is necessary and sufficient for (M,g) to be isometric to the sphere S2m+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.

A Note on Killing Calculus on Riemannian Manifolds

Mathematics ◽

10.3390/math9040307 ◽

2021 ◽

Vol 9 (4) ◽

pp. 307

Author(s):

Sharief Deshmukh ◽

Amira Ishan ◽

Suha B. Al-Shaikh ◽

Cihan Özgür

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Laplace Operator ◽

Compact Riemannian Manifold ◽

Killing Vector ◽

Adjoint Operator ◽

Dimensional Sphere ◽

Killing Vector Field ◽

Smooth Functions ◽

The Laplace Operator

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∞(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 Hh(ξ,ξ) of a smooth function h∈C∞(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.

On Jacobi-Type Vector Fields on Riemannian Manifolds

Mathematics ◽

10.3390/math7121139 ◽

2019 ◽

Vol 7 (12) ◽

pp. 1139 ◽

Cited By ~ 2

Author(s):

Bang-Yen Chen ◽

Sharief Deshmukh ◽

Amira A. Ishan

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Riemannian Manifolds ◽

Vector Fields ◽

Compact Riemannian Manifold ◽

Killing Vector ◽

Sufficient Conditions ◽

Killing Vector Field ◽

Natural Question ◽

Necessary And Sufficient

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.

On an Anti-Torqued Vector Field on Riemannian Manifolds

Mathematics ◽

10.3390/math9182201 ◽

2021 ◽

Vol 9 (18) ◽

pp. 2201

Author(s):

Sharief Deshmukh ◽

Ibrahim Al-Dayel ◽

Devaraja Mallesha Naik

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Riemannian Manifolds ◽

Vector Fields ◽

Compact Riemannian Manifold ◽

Killing Vector ◽

Dual Vector ◽

Definition Of ◽

Simply Connected

A torqued vector field ξ is a torse-forming vector field on a Riemannian manifold that is orthogonal to the dual vector field of the 1-form in the definition of torse-forming vector field. In this paper, we introduce an anti-torqued vector field which is opposite to torqued vector field in the sense it is parallel to the dual vector field to the 1-form in the definition of torse-forming vector fields. It is interesting to note that anti-torqued vector fields do not reduce to concircular vector fields nor to Killing vector fields and thus, give a unique class among the classes of special vector fields on Riemannian manifolds. These vector fields do not exist on compact and simply connected Riemannian manifolds. We use anti-torqued vector fields to find two characterizations of Euclidean spaces. Furthermore, a characterization of an Einstein manifold is obtained using the combination of a torqued vector field and Fischer–Marsden equation. We also find a condition under which the scalar curvature of a compact Riemannian manifold admitting an anti-torqued vector field is strictly negative.

Torsional rigidity on compact Riemannian manifolds with lower Ricci curvature bounds

Open Mathematics ◽

10.1515/math-2015-0053 ◽

2015 ◽

Vol 13 (1) ◽

Author(s):

Najoua Gamara ◽

Abdelhalim Hasnaoui ◽

Akrem Makni

Keyword(s):

Riemannian Manifold ◽

Dirichlet Problem ◽

Ricci Curvature ◽

Riemannian Manifolds ◽

Compact Riemannian Manifold ◽

Torsional Rigidity ◽

Hölder Inequality ◽

Reverse Hölder Inequality ◽

Curvature Bounds ◽

Reverse Hölder

AbstractIn this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains

Twistor Bundle of a Neutral Kähler Surface

Symmetry ◽

10.3390/sym14010043 ◽

2021 ◽

Vol 14 (1) ◽

pp. 43

Author(s):

Włodzimierz Jelonek

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Complex Structure ◽

Killing Vector ◽

Kähler Manifold ◽

Killing Vector Field ◽

Kahler Manifold ◽

Kähler Surface ◽

Kähler Surfaces ◽

Twistor Bundle

In this paper, we characterize neutral Kähler surfaces in terms of their positive twistor bundle. We prove that an O+,+(2,2)-oriented four-dimensional neutral semi-Riemannian manifold (M,g) admits a complex structure J with ΩJ∈⋀−M, such that (M,g,J) is a neutral-Kähler manifold if and only if the twistor bundle (Z1(M),gc) admits a vertical Killing vector field.

Mean curvature flow in a Riemannian manifold endowed with a Killing vector field

Pacific Journal of Mathematics ◽

10.2140/pjm.2020.308.435 ◽

2020 ◽

Vol 308 (2) ◽

pp. 435-472

Author(s):

Liangjun Weng

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Mean Curvature ◽

Mean Curvature Flow ◽

Killing Vector ◽

Curvature Flow ◽

Killing Vector Field

Bernstein-type Theorems in a Riemannian Manifold with an Irrotational Killing Vector Field

Mediterranean Journal of Mathematics ◽

10.1007/s00009-015-0546-y ◽

2015 ◽

Vol 13 (3) ◽

pp. 1285-1290 ◽

Cited By ~ 4

Author(s):

Alfonso Romero ◽

Rafael M. Rubio

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Killing Vector ◽

Killing Vector Field ◽

Bernstein Type

On the Differential Equation Governing Torqued Vector Fields on a Riemannian Manifold

Symmetry ◽

10.3390/sym12121941 ◽

2020 ◽

Vol 12 (12) ◽

pp. 1941

Author(s):

Sharief Deshmukh ◽

Nasser Bin Turki ◽

Haila Alodan

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Euclidean Space ◽

Riemannian Manifolds ◽

Vector Fields ◽

Compact Riemannian Manifold ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Rigidity Results ◽

Necessary And Sufficient

In this article, we show that the presence of a torqued vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian manifolds of constant curvature. More precisely, we show that there is no torqued vector field on n-sphere Sn(c). A nontrivial example of torqued vector field is constructed on an open subset of the Euclidean space En whose torqued function and torqued form are nowhere zero. It is shown that owing to topology of the Euclidean space En, this type of torqued vector fields could not be extended globally to En. Finally, we find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field.

On Affine Transformations of a Riemannian Manifold

Nagoya Mathematical Journal ◽

10.1017/s0027763000023321 ◽

1955 ◽

Vol 9 ◽

pp. 99-109 ◽

Cited By ~ 14

Author(s):

Jun-Ichi Hano

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Affine Transformation ◽

Integral Formula ◽

Killing Vector ◽

Decomposition Theorem ◽

Complete Riemannian Manifold ◽

Killing Vector Field ◽

Affine Transformations ◽

Infinitesimal Affine Transformation

In this paper we establish some theorems about the group of affine transformations on a Riemannian manifold. First we prove a decomposition theorem (Theorem 1) of the largest connected group of affine transformations on a simply connected complete Riemannian manifold, which corresponds to the decomposition theorem of de Rham [4] for the manifold. In the case of the largest group of isometries, a theorem of the same type is found in de Rham’s paper [4] in a weaker form. Using Theorem 1 we obtain a sufficient condition for an infinitesimal affine transformation to be a Killing vector field (Theorem 2). This result includes K. Yano’s theorem [13] which states that on a compact Riemannian manifold an infinitesimal affine transformation is always a Killing vector field. His proof of the theorem depends on an integral formula which is valid only for a compact manifold. Our method is quite different and is based on a result [11] of K. Nomizu.

Fixed Points of Isometries

Nagoya Mathematical Journal ◽

10.1017/s0027763000023497 ◽

1958 ◽

Vol 13 ◽

pp. 63-68 ◽

Cited By ~ 55

Author(s):

Shoshichi Kobayashi

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Fixed Points ◽

Killing Vector ◽

Connected Components ◽

Killing Vector Field ◽

Set Of Points ◽

Infinitesimal Isometry

The purpose of this paper is to prove the followingTheorem. Let M be a Riemannian manifold of dimension n and let ξ be a Killing vector field (i.e., infinitesimal isometry) of M. Let F be the set of points x of M where ξ vanishes and let F = ∪ Vi, where the Vi’s are the connected components of F. Then (assuming F to be non-empty)