η-Ricci solitons and almost η-Ricci solitons on para-Sasakian manifolds
06 Sep 2019-International Journal of Geometric Methods in Modern Physics (World Scientific Publishing Company)-Vol. 16, Iss: 09, pp 1950134
TL;DR: In this paper, the authors studied a para-Sakian manifold whose metric g is an η-Ricci soliton (g,V ) and almost η Ricci solitons.
Abstract: In this paper, we study para-Sasakian manifold (M,g) whose metric g is an η-Ricci soliton (g,V ) and almost η-Ricci soliton. We prove that, if g is an η-Ricci soliton, then either M is Einstein and...
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TL;DR: In this paper , it was shown that if an η \eta -Einstein para-Kenmotsu manifold admits a conformal Ricci almost soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein.
Abstract: Abstract We prove that if an η \eta -Einstein para-Kenmotsu manifold admits a conformal η \eta -Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a conformal η \eta -Ricci soliton is Einstein if its potential vector field V V is infinitesimal paracontact transformation or collinear with the Reeb vector field. Furthermore, we prove that if a para-Kenmotsu manifold admits a gradient conformal η \eta -Ricci almost 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 conformal η \eta -Ricci soliton and satisfy our results. We also have studied conformal η \eta -Ricci soliton in three-dimensional para-cosymplectic manifolds.
20 citations
TL;DR: In this article, the authors considered the problem of paracontact geometry on a para-Kenmotsu manifold and showed that if the metric g of g of G of σ, σ is a Gaussian, then G is either the potential vector field collinear with Reeb vector field or Ricci soliton.
Abstract: We consider almost
$$*$$
-Ricci solitons in the context of paracontact geometry, precisely, on a paraKenmotsu manifold. First, we prove that if the metric g of
$$\eta $$
-Einstein paraKenmotsu manifold is
$$*$$
Ricci soliton, then M is Einstein. Next, we show that if
$$\eta $$
-Einstein paraKenmotsu manifold admits a gradient almost
$$*$$
-Ricci soliton, then either M is Einstein or the potential vector field collinear with Reeb vector field
$$\xi $$
. Finally, for three-dimensional case we show that paraKenmotsu manifold is of constant curvature
$$-1$$
. An illustrative example is given to support the obtained results.
12 citations
25 Sep 2020
10 citations
TL;DR: In this paper, the authors studied the Riemann soliton and gradient almost-Riemann-soliton on a certain class of almost Kenmotsu manifolds.
Abstract: The aim of this paper, is to study the Riemann soliton and gradient almost Riemann soliton on certain class of almost Kenmotsu manifolds. Also, some suitable examples of Kenmotsu and (κ,μ)′-almost ...
10 citations
11 May 2021
TL;DR: In this paper, the authors characterized the Einstein metrics in such broad classes of metrics as almost $$\eta $$¯¯ -Ricci solitons and almost $€  ¯¯¯¯ -RICci soliton on Kenmotsu manifolds, and generalized some known results.
Abstract: We characterize the Einstein metrics in such broad classes of metrics as almost $$\eta $$
-Ricci solitons and $$\eta $$
-Ricci solitons on Kenmotsu manifolds, and generalize some known results. First, we prove that a Kenmotsu metric as an $$\eta $$
-Ricci soliton is Einstein metric if either it is $$\eta $$
-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 $$\eta $$
-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of $$\eta $$
-Ricci solitons and gradient $$\eta $$
-Ricci solitons, which illustrate our results.
9 citations
Cites methods from "η-Ricci solitons and almost η-Ricci..."
...[2, 3, 4] and Naik-Venkatesha [24]....
[...]
...Many authors studied Ricci solitons, η-Ricci solitons and their generalizations in the framework of almost contact and paracontact geometries, e.g., contact metrics as Ricci solitons by Cho-Sharma [9], K-contact and (k, µ)-contact metrics as Ricci solitons by Sharma [27], contact metrics as Ricci almost solitons by Ghosh-Sharma [16, 28], contact metrics as h-almost Ricci solitons by Ghosh-Patra [19], almost contact B-metrics as Ricci-like soliton by Manev [23], para-Sasakian and Lorentzian para-Sasakian metrics as η-Ricci solitons by Naik-Venkatesha [24] and Blaga [3], etc. Based on the above results in a modern and active field of research, a natural question can be posed: Are there almost contact metric manifolds, whose metrics are η-Ricci solitons?...
[...]
..., contact metrics as Ricci solitons by Cho-Sharma [9], K-contact and (k, μ)-contact metrics as Ricci solitons by Sharma [27], contact metrics as Ricci almost solitons by Ghosh-Sharma [16, 28], contact metrics as h-almost Ricci solitons by Ghosh-Patra [19], almost contact B-metrics as Ricci-like soliton by Manev [23], para-Sasakian and Lorentzian para-Sasakian metrics as η-Ricci solitons by Naik-Venkatesha [24] and Blaga [3], etc....
[...]
References
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08 Jan 2002
TL;DR: In this article, the authors describe a complex geometry model of Symplectic Manifolds with principal S1-bundles and Tangent Sphere Bundles, as well as a negative Xi-sectional Curvature.
Abstract: Preface * 1. Symplectic Manifolds * 2. Principal S1-bundles * 3. Contact Manifolds * 4. Associated Metrics * 5. Integral Submanifolds and Contact Transformations * 6. Sasakian and Cosymplectic Manifolds * 7. Curvature of Contact Metric Manifolds * 8. Submanifolds of Kahler and Sasakian Manifolds * 9. Tangent Bundles and Tangent Sphere Bundles * 10. Curvature Functionals and Spaces of Associated Metrics * 11. Negative Xi-sectional Curvature * 12. Complex Contact Manifolds * 13. Additional Topics in Complex Geometry * 14. 3-Sasakian Manifolds * Bibliography * Subject Index * Author Index
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24 May 2004TL;DR: The Ricci flow of special geometries Special and limit solutions Short time existence Maximum principles The Ricci Flow on surfaces Three-manifolds of positive Ricci curvature Derivative estimates Singularities and the limits of their dilations Type I singularities as discussed by the authors.
Abstract: The Ricci flow of special geometries Special and limit solutions Short time existence Maximum principles The Ricci flow on surfaces Three-manifolds of positive Ricci curvature Derivative estimates Singularities and the limits of their dilations Type I singularities The Ricci calculus Some results in comparison geometry Bibliography Index.
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TL;DR: In this article it was shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the structure is skew symmetric and the defining vector field is Killing.
Abstract: The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A $${\mathcal{D}}$$
-homothetic transformation is determined as a special gauge transformation. The η-Einstein manifold are defined, it is proved that their scalar curvature is a constant, and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with $${\mathcal{D}}$$
-homothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skew-symmetric and the defining vector field is Killing.
229 citations
TL;DR: In this article, Boyer and Galicki showed that a complete K-contact gradient soliton is a Jacobi vector field along the geodesics of the Reeb vector field.
Abstract: Inspired by a result of Boyer and Galicki, we prove that a complete K-contact gradient soliton is compact Einstein and Sasakian. For the non-gradient case we show that the soliton vector field is a Jacobi vector field along the geodesics of the Reeb vector field. Next we show that among all complete and simply connected K-contact manifolds only the unit sphere admits a non-Killing holomorphically planar conformal vector field (HPCV). Finally we show that, if a (k, μ)-contact manifold admits a non-zero HPCV, then it is either Sasakian or locally isometric to E3 or En+1 × Sn (4).
157 citations
TL;DR: In this article, it was shown that a real hypersurface in a non-flat complex space form does not admit a Ricci soliton whose potential vector field is the Reeb vector field.
Abstract: We prove that a real hypersurface in a non-flat complex space form does not admit a Ricci soliton whose potential vector field is the Reeb vector field. Moreover, we classify a real hypersurface admitting so-called “$\eta$-Ricci soliton” in a non-flat complex space form.
154 citations