Some results on almost Ricci solitons and geodesic vector fields

Sharma, Ramesh

doi:10.1007/s13366-017-0367-1

Cited by 13 publications

(4 citation statements)

References 12 publications

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“…One of the themes of interest in Riemannian geometry concerns the presence of some vector fields of special type on Riemannian spaces, like (unit) geodesic vector fields (cf. [1][2][3]), Jacobi-type vector fields (cf. [4,5]), concircular vector fields (cf.…”

Section: Introductionmentioning

confidence: 99%

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: 99%

A Note on Geodesic Vector Fields

et al. 2020

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“…Furthermore, Riemann's soliton concerning infinitesimal harmonic transformation was studied in [18]. In this association, we notice that Sharma in [16] explored almost Ricci soliton in K-contact geometry and in [17], with divergence-free soliton vector field. In [6], Riemann soliton under the context of contact manifold has been studied and demonstrated a few intriguing outcomes.…”

Section: Introductionmentioning

confidence: 99%

A note on Almost Riemann Soliton and gradient almost Riemann soliton

De¹,

De²

2020

Preprint

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The quest of the offering article is to investigate almost Riemann soliton and gradient almost Riemann soliton in a non-cosymplectic normal almost contact metric manifold M 3 . Before all else, it is proved that if the metric of M 3 is Riemann soliton with divergence-free potential vector field Z, then the manifold is quasi-Sasakian and is of constant sectional curvature -λ, provided α, β = constant. Other than this, it is shown that if the metric of M 3 is ARS and Z is pointwise collinear with ξ and has constant divergence, then Z is a constant multiple of ξ and the ARS reduces to a Riemann soliton, provided α, β =constant. Additionally, it is established that if M 3 with α, β = constant admits a gradient ARS (γ, ξ, λ), then the manifold is either quasi-Sasakian or is of constant sectional curvature −(α 2 − β 2 ). At long last, we develop an example of M 3 conceding a Riemann soliton.

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“…[3]) in an attempt to generalize Ricci solitons, by replacing the soliton constant with the smooth function σ. Geometry of Ricci solitons and Ricci almost solitons has been subject of immense interest due to their elegant geometry as well as applications (cf. [1,[3][4][5][6][7][8][9][10][11][12][13][14][15]). Given a Ricci almost soliton (M, g, w, σ), we call w the soliton vector field and the smooth function σ the potential function.…”

Section: Introductionmentioning

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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Some results on almost Ricci solitons and geodesic vector fields

Cited by 13 publications

References 12 publications

A Note on Geodesic Vector Fields

A Note on Geodesic Vector Fields

A note on Almost Riemann Soliton and gradient almost Riemann soliton

Some Results on Ricci Almost Solitons

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