Geodesic Vector Fields on a Riemannian Manifold

Deshmukh, Sharief; Peška, Patrik; Turki, Nasser Bin

doi:10.3390/math8010137

Cited by 14 publications

(10 citation statements)

References 25 publications

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“…Theorem 1). It is noteworthy that some distinct characterizations of the sphere and the Euclidean space in terms of the existence of non-trivial geodesic vector fields satisfying certain additional properties were recently obtained in [31], but these are of different nature and do not imply the characterizations established in the present work (cf. Remarks 1 and 2).…”

Section: Introductionmentioning

confidence: 60%

A Note on Geodesic Vector Fields

et al. 2020

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The concircularity property of vector fields implies the geodesicity property, while the converse of this statement is not true. The main objective of this note is to find conditions under which the concircularity and geodesicity properties of vector fields are equivalent. Moreover, it is shown that the geodesicity property of vector fields is also useful in characterizing not only spheres, but also Euclidean spaces.

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Section: Introductionmentioning

confidence: 60%

A Note on Geodesic Vector Fields

et al. 2020

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“…Combining Equation (26) with Equation (24), we conclude (n − 2) (∇σ − σv) = 0 and as n > 0, we have ∇σ = σv. (27) Note that owing to the conditions on the torqued function σ and the torqued form α, the vector field σv is nowhere zero on the Euclidean space E n .…”

Section: Theoremmentioning

confidence: 85%

“…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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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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“…Most importantly, on all Kcontact manifolds there is a unit Killing vector field called the Reeb vector field (see [12,13]). There are other important structures and special vector fields, which also influence the geometry of a Riemannian manifold (see [14]).…”

Section: Introductionmentioning

confidence: 99%

On Killing Vector Fields on Riemannian Manifolds

Deshmukh

Belova

2021

Mathematics

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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.

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Geodesic Vector Fields on a Riemannian Manifold

Cited by 14 publications

References 25 publications

A Note on Geodesic Vector Fields

A Note on Geodesic Vector Fields

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

On Killing Vector Fields on Riemannian Manifolds

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