A Note on Geodesic Vector Fields

Deshmukh, Sharief; Mikeš, Josef; Turki, Nasser Bin; Vîlcu, Gabriel-Eduard

doi:10.3390/math8101663

Cited by 6 publications

(5 citation statements)

References 30 publications

(35 reference statements)

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“…It is interesting to observe that the above warped product is short of being a twisted product and thus our Theorem 1 strengthens that the conclusion of Theorem 3.1 in [25] is sharp.…”

Section: Torqued Vector Fields On Spheres and Euclidean Spacessupporting

confidence: 72%

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

2020

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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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“…It is interesting to observe that the above warped product is short of being a twisted product and thus our Theorem 1 strengthens that the conclusion of Theorem 3.1 in [25] is sharp.…”

Section: Torqued Vector Fields On Spheres and Euclidean Spacessupporting

confidence: 72%

“…Most basic among special vector fields are geodesic vector fields. In [9,25,26] it has been shown that geodesic vector fields are useful in characterizing spheres and Euclidean spaces.…”

Section: Introductionmentioning

confidence: 99%

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

2020

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“…There are other types of vector fields, for instance generalized geodesic vector fields, which are closely related to conformal vector fields (cf. [20]), it will be interesting to investigate the role of generalized geodesic vector fields on compact Ricci almost solitons in making them isometric to the sphere S n (c).…”

Section: Discussionmentioning

confidence: 99%

Some Results on Ricci Almost Solitons

2021

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We find three necessary and sufficient conditions for an n-dimensional compact Ricci almost soliton (M,g,w,σ) to be a trivial Ricci soliton under the assumption that the soliton vector field w is a geodesic vector field (a vector field with integral curves geodesics). The first result uses condition r2≤nσr on a nonzero scalar curvature r; the second result uses the condition that the soliton vector field w is an eigen vector of the Ricci operator with constant eigenvalue λ satisfying n2λ2≥r2; the third result uses a suitable lower bound on the Ricci curvature S(w,w). Finally, we show that an n-dimensional connected Ricci almost soliton (M,g,w,σ) with soliton vector field w is a geodesic vector field with a trivial Ricci soliton, if and only if, nσ−r is a constant along integral curves of w and the Ricci curvature S(w,w) has a suitable lower bound.

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“…Riemannian manifolds with Killing vector fields has been subject of interest for many mathematicians (cf. [2,[4][5][6][7][8][9][10][11][12][13]). There are other important vector fields, such as Jacobi-type vector fields, geodesic vector fields and torqued vector fields, which play important roles in the geometry of a Riemannian manifold (cf.…”

Section: Introductionmentioning

confidence: 99%

“…There are other important vector fields, such as Jacobi-type vector fields, geodesic vector fields and torqued vector fields, which play important roles in the geometry of a Riemannian manifold (cf. [10,11,[14][15][16]). Moreover, incompressible vector fields have applications in Physics, and as Killing vector fields are incompressible, they have applications in Physics (cf.…”

Section: Introductionmentioning

confidence: 99%

A Note on Killing Calculus on Riemannian Manifolds

et al. 2021

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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.

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A Note on Geodesic Vector Fields

Cited by 6 publications

References 30 publications

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

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

Some Results on Ricci Almost Solitons

A Note on Killing Calculus on Riemannian Manifolds

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