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.

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.

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.

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 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 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 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.

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.

Frictional magnetic curves in 3D Riemannian manifolds

2018 ◽  
Vol 15 (02) ◽  
pp. 1850020 ◽  
Author(s):  
Talat Korpinar ◽  
Ridvan Cem Demirkol
Keyword(s):  
Magnetic Field ◽  
Riemannian Manifold ◽  
Charged Particle ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Frictional Force ◽  
Magnetic Vector ◽  
Geometrical Properties ◽  
Physical Consequences ◽  
The Magnetic Field

In this study, we investigate the special type of magnetic trajectories associated with a magnetic field [Formula: see text] defined on a 3D Riemannian manifold. First, we consider a moving charged particle under the action of a frictional force, [Formula: see text], in the magnetic field [Formula: see text]. Then, we assume that trajectories of the particle associated with the magnetic field [Formula: see text] correspond to frictional magnetic curves ([Formula: see text]-magnetic curves[Formula: see text] of magnetic vector field [Formula: see text] on the 3D Riemannian manifold. Thus, we are able to investigate some geometrical properties and physical consequences of the particle under the action of frictional force in the magnetic field [Formula: see text] on the 3D Riemannian manifold.

scholarly journals On Killing Vector Fields on Riemannian Manifolds

Mathematics ◽  
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.

scholarly journals $L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Li ◽  
Shuxiang Feng ◽  
Peibiao Zhao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Finiteness Theorem ◽  
Conformally Flat ◽  
Locally Conformally Flat ◽  
Flat Riemannian Manifold ◽  
Ricci Form

AbstractIn this paper, we establish a finiteness theorem for $L^{p}$ L p harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on $L^{2}$ L 2 harmonic 1-forms.

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