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Journal Article•DOI•

Riemann Soliton within the framework of contact geometry

M.N. Devaraja1, H. Aruna Kumara1, V. Venkatesha1•
Kuvempu University1
04 May 2021-Quaestiones Mathematicae (National Inquiry Services Center (NISC))-Vol. 44, Iss: 5, pp 637–651-637–651
TL;DR: In this paper, a contact metric manifold whose metric is a Riemann soliton was studied and it was shown that the manifold is either of constant curvature + 1 (and V is Killing) or D-homothetically invariant.
Abstract: In this paper, we study contact metric manifold whose metric is a Riemann soliton. First, we consider Riemann soliton (g; V ) with V as contact vector eld on a Sasakian manifold (M; g) and in this case we prove that M is either of constant curvature +1 (and V is Killing) or D-homothetically xed -Einstein manifold (and V leaves the structure tensor φ invariant). Next, we prove that if a compact K-contact manifold whose metric g is a gradient almost Riemann soliton, then it is Sasakian and isometric to a unit sphere S2n+1. Further, we study H-contact manifold admitting a Riemann soliton (g; V ) where V is pointwise collinear with .Key words: Contact metric manifold, Riemann soliton, gradient almost Riemann soliton.
Citations
More filters
Journal Article•
Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

[...]

Morio Obata
01 Jan 1962-Tokyo Sugaku Kaisya Zasshi

511 citations

Book•
Integral formulas in Riemannian geometry

[...]

健太郎 矢野
01 Jan 1970

329 citations

Journal Article•DOI•
Riemann solitons and almost Riemann solitons on almost Kenmotsu manifolds

[...]

V. Venkatesha1, H. Aruna Kumara1, Devaraja Mallesha Naik1•
Kuvempu University1
18 Jun 2020-International Journal of Geometric Methods in Modern Physics
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

Cites result from "Riemann Soliton within the framewor..."

  • ...To fulfill this classification, the present authors in [4] consider a Riemann soliton on contact manifold, and obtained several intersting results....

    [...]

Posted Content•
A note on Almost Riemann Soliton and gradient almost Riemann soliton.

[...]

Krishnendu De, Uday Chand De
24 Aug 2020-arXiv: Differential Geometry
TL;DR: In this paper, it was shown that if the metric of a non-cosymplectic normal almost contact metric manifold is Riemann soliton with divergence-free potential vector field (Z), then the manifold is quasi-Sakian and is of constant sectional curvature -$\lambda.
Abstract: The quest of the offering article is to investigate \emph{almost Riemann soliton} and \emph{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 -$\lambda$, provided $\alpha,\beta =$ constant. Other than this, it is shown that if the metric of $M^3$ is \emph{ARS} and $Z$ is pointwise collinear with $\xi $ and has constant divergence, then $Z$ is a constant multiple of $\xi $ and the \emph{ARS} reduces to a Riemann soliton, provided $\alpha,\;\beta =$constant. Additionally, it is established that if $M^3$ with $\alpha,\; \beta =$ constant admits a gradient \emph{ARS} $(\gamma,\xi,\lambda)$, then the manifold is either quasi-Sasakian or is of constant sectional curvature $-(\alpha^2-\beta^2)$. At long last, we develop an example of $M^3$ conceding a Riemann soliton.

7 citations

Cites background from "Riemann Soliton within the framewor..."

  • ...8 of [6]) For any vector fields E,F on M(3), in a gradient ARS (M,g, γ,m, λ), we infer R(E,F )Dγ = (∇FQ)E − (∇EQ)F +{F (2λ+△γ)E − E(2λ+△γ)F}, (5....

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  • ...In [6], Riemann soliton under the context of contact manifold has been studied and demonstrated a few intriguing outcomes....

    [...]

Posted Content•
Remarks on almost Riemann solitons with gradient or torse-forming vector field

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Adara M. Blaga1•
West University of Timișoara1
14 Aug 2020-arXiv: Differential Geometry
TL;DR: In this article, almost Riemann solitons were considered in a manifold and underlined their relation to almost Ricci (Riemannian) soliton and Ricci solitón.
Abstract: We consider almost Riemann solitons $(V,\lambda)$ in a Riemannian manifold and underline their relation to almost Ricci solitons. When $V$ is of gradient type, using Bochner formula, we explicitly express the function $\lambda$ by means of the gradient vector field $V$ and illustrate the result with suitable examples. Moreover, we deduce some properties for the particular cases when the potential vector field of the soliton is solenoidal or torse-forming, with a special view towards curvature.

6 citations

References
More filters
Book•
Riemannian Geometry of Contact and Symplectic Manifolds

[...]

David E. Blair
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

1,822 citations

"Riemann Soliton within the framewor..." refers background in this paper

  • ...Consider the unit sphere S ⊂ R and this admits a standard Sasakian structure (φ, ξ, η, g) [2]....

    [...]

Journal Article•
Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

[...]

Morio Obata
01 Jan 1962-Tokyo Sugaku Kaisya Zasshi

511 citations

"Riemann Soliton within the framewor..." refers methods in this paper

  • ...We apply the celebrated Obata’s theorem [10] to finish the proof of the theorem....

    [...]

  • ...We invoke the celebrated Obata’s theorem [10] to find a non-trivial smooth function ν such that ∇∇ν = −νg....

    [...]

Book•
Integral formulas in Riemannian geometry

[...]

健太郎 矢野
01 Jan 1970

329 citations

Additional excerpts

  • ...Again from [20], we know that £VR k hji = ∇h(£V ∇)ij −∇j(£V ∇)ih....

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  • ...The following formula is well-known (see [20])...

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Journal Article•DOI•
The topology of contact Riemannian manifolds

[...]

Shûkichi Tanno1•
Tohoku University1
01 Dec 1968-Illinois Journal of Mathematics

217 citations

"Riemann Soliton within the framewor..." refers background in this paper

  • ...D-homothetic deformation was introduced by Tanno [17] which is the deformed structure φ̄ = φ, ḡ = ag + a(a− 1)η ⊗ η, , η̄ = aη, ξ̄ = 1 a ξ, where a ∈ R+....

    [...]

  • ...S. Tanno, The topology of contact Riemannian manifolds, Ill. J. Math....

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  • ...D-homothetic deformation was introduced by Tanno [17] which is the deformed structure...

    [...]

Journal Article•DOI•
Certain Results on K-Contact and (k, μ)-Contact Manifolds

[...]

Ramesh Sharma1•
University of New Haven1
22 Sep 2008-Journal of Geometry
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

"Riemann Soliton within the framewor..." refers background in this paper

  • ...Sharma in [13] extended this result by proving that if a complete K-contact metric is a gradient Ricci soliton, then it is a compact Sasaki-Einstein manifold....

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