Almost $η$-Ricci and almost $η$-Yamabe solitons with torse-forming potential vector field

Blaga, Adara M.; Özgür, Cıhan

doi:10.48550/arxiv.2003.12574

Cited by 4 publications

(6 citation statements)

References 11 publications

(17 reference statements)

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“…In [38], Welyczko introduced notion of para-Kenmotsu manifold, which is the analogous of Kenmotsu manifold [19] in paracontact geometry and detailed studied by Zamkovoy [42]. Further, Blaga studied some aspects of η-Ricci solitons on para-Kenmotsu and Lorentzian para-Sasakian manifolds (see [4,5,6]). Recently, Patra [23] consider Ricci soliton on para-Kenmotsu manifold and proved that a para-Kenmotsu metric as a Ricci soliton is Einstein if it is η-Einstein or the potential vector field V is infinitesimal paracontact transformation.…”

Section: Introductionmentioning

confidence: 99%

CERTAIN RESULTS ON $\eta$-RICCI SOLITIONS AND ALMOST $\eta$-RICCI SOLITONS

Dey¹,

Azami²

2022

FU Math Inform

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We prove that if an $\eta$-Einstein para-Kenmotsu manifold admits a $\eta$-Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a $\eta$-Ricci soliton is Einstein if its potential vector field $V$ is infinitesimal paracontact transformation or collinear with the Reeb vector field. Further, we prove that if a para-Kenmotsu manifold admits a gradient almost $\eta$-Ricci soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits $\eta$-Ricci soliton and satisfy our results. We also have studied $\eta$-Ricci soliton in 3-dimensional normal almost paracontact metric manifolds and we show that if in a 3-dimensional normal almost paracontact metric manifold with $\alpha, \beta $ = constant, the metric is $\eta$-Ricci soliton, where potential vector ﬁeld $V$ is collinear with the characteristic vector ﬁeld $\xi$, then the manifold is $\eta$-Einstein manifold.

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Section: Introductionmentioning

confidence: 99%

CERTAIN RESULTS ON $\eta$-RICCI SOLITIONS AND ALMOST $\eta$-RICCI SOLITONS

Dey¹,

Azami²

2022

FU Math Inform

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“…al. [2,3,4] and Naik-Venkatesha [24]. An η-Ricci soliton is said to be almost η-Ricci soliton if λ and µ are smooth functions on M (see details in [2,4]).…”

Section: Introductionmentioning

confidence: 99%

“…[2,3,4] and Naik-Venkatesha [24]. An η-Ricci soliton is said to be almost η-Ricci soliton if λ and µ are smooth functions on M (see details in [2,4]). When the potential vector field V is a gradient of a smooth function f : M → R (called the potential function) the manifold will be called a gradient almost η-Ricci soliton, and (3) reads as…”

Section: Introductionmentioning

confidence: 99%

Almost $η$-Ricci solitons on Kenmotsu manifolds

Patra¹,

Rovenski²

2020

Preprint

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In this paper we characterize the Einstein metrics in such broader classes of metrics as almost η-Ricci solitons and η-Ricci solitons on Kenmotsu manifolds, and generalize some results of other authors. First, we prove that a Kenmotsu metric as an η-Ricci soliton is Einstein metric if either it is η-Einstein or the potential vector field V is an infinitesimal contact transformation or V is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost η-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of η-Ricci solitons and gradient η-Ricci solitons, which illustrate our results.

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“…Although Ricci solitons were first studied in Riemannian geometry, they and their generalizations have recently been intensively studied for pseudo-Riemannian metrics, mainly on Lorentzian and paracontact metric manifolds (e.g. [4], [5], [6], [7], [12], [30], [33], [39], [40]).…”

Section: Introductionmentioning

confidence: 99%

“…Their research has recently been expanded to include mainly η-Ricci solitons with torse-forming potential that is orthogonal to ker η and the solitons are compatible with various additional tensor structures (e.g. [5], [6], [7], [39], [40]).…”

Section: Introductionmentioning

confidence: 99%

Almost Ricci-like solitons with torse-forming vertical potential of constant length on almost contact B-metric manifolds

Manev

2020

Preprint

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Almost Ricci-like solitons on almost contact B-metric manifolds with torse-forming potential have been studied. The case in which this potential is further vertical is considered, i.e. the potential is pointwise collinear to the Reeb vector field. The conditions under which the investigated manifolds with almost Ricci-like solitons are equivalent to almost Einstein-like manifolds have been established. In this case, necessary and sufficient conditions have been found for a number of properties of the curvature tensor and its Ricci tensor. Then some results are obtained for a parallel symmetric second-order covariant tensor on the manifolds under study. Finally, an explicit example of an arbitrary dimension is given and some of the results are illustrated.

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Almost $η$-Ricci and almost $η$-Yamabe solitons with torse-forming potential vector field

Cited by 4 publications

References 11 publications

CERTAIN RESULTS ON $\eta$-RICCI SOLITIONS AND ALMOST $\eta$-RICCI SOLITONS

CERTAIN RESULTS ON $\eta$-RICCI SOLITIONS AND ALMOST $\eta$-RICCI SOLITONS

Almost $η$-Ricci solitons on Kenmotsu manifolds

Almost Ricci-like solitons with torse-forming vertical potential of constant length on almost contact B-metric manifolds

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