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

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.

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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.

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Killing Vector Fields in Generalized Conformalβ-Change of Finsler Spaces

Journal of Mathematics ◽

10.1155/2015/456291 ◽

2015 ◽

Vol 2015 ◽

pp. 1-5

Author(s):

Mallikarjun Yallappa Kumbar ◽

Narasimhamurthy Senajji Kampalappa ◽

Thippeswamy Komalobiah Rajanna ◽

Kavyashree Ambale Rajegowda

Keyword(s):

Vector Field ◽

Vector Fields ◽

Finsler Space ◽

Killing Vector ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Finsler Spaces ◽

Killing Vector Fields ◽

Necessary And Sufficient

We consider a Finsler space equipped with a Generalized Conformalβ-change of metric and study the Killing vector fields that correspond between the original Finsler space and the Finsler space equipped with Generalized Conformalβ-change of metric. We obtain necessary and sufficient condition for a vector field Killing in the original Finsler space to be Killing in the Finsler space equipped with Generalized Conformalβ-change of metric.

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Anti-invariant Riemanian submersions from nearly-k-cosymplectic manifolds

Filomat ◽

10.2298/fil1804159t ◽

2018 ◽

Vol 32 (4) ◽

pp. 1159-1174

Author(s):

Ju Tan ◽

Na Xu

Keyword(s):

Vector Field ◽

Riemannian Manifolds ◽

Riemannian Submersion ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Riemannian Submersions ◽

Totally Geodesic ◽

Cosymplectic Manifolds ◽

Necessary And Sufficient ◽

Characteristic Vector Field

In this paper, we introduce anti-invariant Riemannian submersions from nearly-K-cosymplectic manifolds onto Riemannian manifolds. We study the integrability of horizontal distributions. And we investigate the necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. Moreover, we give examples of anti-invariant Riemannian submersions such that characteristic vector field ? is vertical or horizontal.

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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.

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Remarks on metallic warped product manifolds

Facta Universitatis Series Mathematics and Informatics ◽

10.22190/fumi1802269b ◽

2018 ◽

Vol 33 (2) ◽

pp. 269

Author(s):

Adara-Monica Blaga ◽

Cristina-Elena Hretcanu

Keyword(s):

Riemannian Manifold ◽

Riemannian Manifolds ◽

Warped Product ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Metallic Structure ◽

Necessary And Sufficient

We characterize the metallic structure on the product of two metallic manifolds in terms of metallic maps and provide a necessary and sufficient condition for the warped product of two locally metallic Riemannian manifolds to be locally metallic. The particular case of product manifolds is discussed and an example of metallic warped product Riemannian manifold is provided.

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Some results about concircular vector fields on Riemannian manifolds

Filomat ◽

10.2298/fil2003835c ◽

2020 ◽

Vol 34 (3) ◽

pp. 835-842

Author(s):

Bang-Yen Chen ◽

Sharief Deshmukh

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Riemannian Manifolds ◽

Vector Fields ◽

Euclidean Spaces ◽

Rigidity Results ◽

Kaehler Manifolds

In this article, we show that the presence of a concircular vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian and Kaehler manifolds. More precisely, we find new geometrical characterizations of spheres, Euclidean spaces as well as of complex Euclidean spaces using non-trivial concircular vector fields.

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Para-Ricci-like Solitons on Almost Paracontact Almost Paracomplex Riemannian Manifolds

10.20944/preprints202106.0622.v1 ◽

2021 ◽

Author(s):

Hristo Manev ◽

Mancho Manev

Keyword(s):

Vector Field ◽

Riemannian Manifolds ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Reeb Vector ◽

Reeb Vector Field ◽

Special Cases ◽

Necessary And Sufficient

It is introduced and studied para-Ricci-like solitons with potential Reeb vector field on almost paracontact almost paracomplex Riemannian manifolds. The special cases of para-Einstein-like, para-Sasaki-like and having a torse-forming Reeb vector field have been considered. It is proved a necessary and sufficient condition the manifold to admit a para-Ricci-like soliton which is the structure to be para-Einstein-like. Explicit examples are provided in support of the proven statements.

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ISOMETRIC IMMERSIONS INTO LORENTZIAN PRODUCTS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887811005695 ◽

2011 ◽

Vol 08 (06) ◽

pp. 1269-1290 ◽

Cited By ~ 5

Author(s):

JULIEN ROTH

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Shape Operator ◽

Equivalent Condition ◽

Sufficient Condition ◽

Maximal Surface ◽

Necessary And Sufficient Condition ◽

Isometric Immersions ◽

Vertical Vector ◽

Necessary And Sufficient

We give a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed into one of the Lorentzian products 𝕊n × ℝ1 or ℍn × ℝ1. This condition is expressed in terms of its first and second fundamental forms, the tangent and normal projections of the vertical vector field. As applications, we give an equivalent condition in a spinorial way and we deduce the existence of a one-parameter family of isometric maximal deformation of a given maximal surface obtained by rotating the shape operator.

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Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

Journal of Geometry and Symmetry in Physics ◽

10.7546/jgsp-62-2021-53-66 ◽

2021 ◽

Vol 62 ◽

pp. 53-66

Author(s):

Fethi Latti ◽

◽

Hichem Elhendi ◽

Lakehal Belarbi

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Tangent Bundle ◽

Vector Fields ◽

Sufficient Conditions ◽

Harmonic Vector ◽

New Class ◽

Necessary And Sufficient ◽

Sasakian Metric ◽

Harmonic Vector Fields

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

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Anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds

Filomat ◽

10.2298/fil1507429m ◽

2015 ◽

Vol 29 (7) ◽

pp. 1429-1444 ◽

Cited By ~ 6

Author(s):

Cengizhan Murathan ◽

Erken Küpeli

Keyword(s):

Riemannian Manifolds ◽

Riemannian Submersion ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Riemannian Submersions ◽

Totally Geodesic ◽

Cosymplectic Manifolds ◽

Necessary And Sufficient ◽

Decomposition Theorems ◽

Characteristic Vector Field

We introduce anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on cosymplectic manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. We give examples of anti-invariant submersions such that characteristic vector field ? is vertical or horizontal. Moreover we give decomposition theorems by using the existence of anti-invariant Riemannian submersions.

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