scholarly journals Some results about concircular vector fields on Riemannian manifolds

Filomat ◽  
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.

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

Symmetry ◽  
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.

scholarly journals Classification of Ricci solitons on Euclidean hypersurfaces

2014 ◽  
Vol 25 (11) ◽  
pp. 1450104 ◽  
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Potential Field ◽  
Vector Fields ◽  
Ricci Soliton ◽  
Position Vector ◽  
Ricci Solitons ◽  
Euclidean Submanifolds

A Ricci soliton (M, g, v, λ) on a Riemannian manifold (M, g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in [Ricci solitons and concurrent vector fields, preprint (2014), arXiv:1407.2790]. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

scholarly journals On Jacobi-Type Vector Fields on Riemannian Manifolds

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1139 ◽  
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.

scholarly journals Geodesic Vector Fields on a Riemannian Manifold

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 137 ◽  
Author(s):  
Sharief Deshmukh ◽  
Patrik Peska ◽  
Nasser Bin Turki
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Vector Fields ◽  
Euclidean Spaces ◽  
Smooth Vector ◽  
Smooth Vector Field ◽  
Geodesic Vector ◽  
Integral Curves

A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of its integral curves is proportional to velocity. In this paper, we show that the presence of a geodesic vector field on a Riemannian manifold influences its geometry. We find characterizations of n-spheres as well as Euclidean spaces using geodesic vector fields.

scholarly journals Closed conformal vector fields on pseudo-Riemannian manifolds

2006 ◽  
Vol 2006 ◽  
pp. 1-8 ◽  
Author(s):  
D. A. Catalano
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Geometric Proof ◽  
Conformal Vector Field ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Local Coordinates

We give here a geometric proof of the existence of certain local coordinates on a pseudo-Riemannian manifold admitting a closed conformal vector field.

scholarly journals Geometry of k-Yamabe Solitons on Euclidean Spaces and Its Applications to Concurrent Vector Fields

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 222
Author(s):  
Akram Ali ◽  
Fatemah Mofarreh ◽  
Pişcoran Laurian-Ioan ◽  
Nadia Alluhaibi
Keyword(s):  
Vector Field ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Point Of View ◽  
Euclidean Spaces ◽  
Field Point ◽  
Yamabe Solitons ◽  
Yamabe Soliton

In this paper, we give some classifications of the k-Yamabe solitons on the hypersurfaces of the Euclidean spaces from the vector field point of view. In several results on k-Yamabe solitons with a concurrent vector field on submanifolds in Riemannian manifolds, is proved that a k-Yamabe soliton (Mn,g,vT,λ) on a hypersurface in the Euclidean space Rn+1 is contained either in a hypersphere or a hyperplane. We provide an example to support this study and all of the results in this paper can be implemented to Yamabe solitons for k-curvature with k=1.

scholarly journals On an Anti-Torqued Vector Field on Riemannian Manifolds

Mathematics ◽  
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.

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

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.

scholarly journals A New Approach on Helices in Pseudo-Riemannian Manifolds

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Evren Zıplar ◽  
Yusuf Yaylı ◽  
İsmail Gök
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Frenet Frame ◽  
New Approach ◽  
Nonzero Constant ◽  
Slant Helix

A proper curveαin then-dimensional pseudo-Riemannian manifold(M,g)is called aVn-slant helix if the functiong(Vn,X)is a nonzero constant alongα, whereXis a  parallel vector field alongαandVnisnth Frenet frame. In this work, we study such curves and give important characterizations about them.

Curvature of K-contact Semi-Riemannian Manifolds

2014 ◽  
Vol 57 (2) ◽  
pp. 401-412 ◽  
Author(s):  
Domenico Perrone
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Lorentzian Manifold ◽  
Constant Sectional Curvature ◽  
Conformally Flat ◽  
Reeb Vector ◽  
Reeb Vector Field

Abstract.In this paper we characterize K-contact semi-Riemannian manifolds and Sasakian semi- Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat K-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature κ = ɛ, where ɛ = ± denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a K-contact Lorentzian manifold.

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