• Title/Summary/Keyword: Ricci soliton

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∗-RICCI SOLITONS AND ∗-GRADIENT RICCI SOLITONS ON 3-DIMENSIONAL TRANS-SASAKIAN MANIFOLDS

  • Dey, Dibakar;Majhi, Pradip
    • Communications of the Korean Mathematical Society
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    • v.35 no.2
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    • pp.625-637
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    • 2020
  • The object of the present paper is to characterize 3-dimensional trans-Sasakian manifolds of type (α, β) admitting ∗-Ricci solitons and ∗-gradient Ricci solitons. Under certain restrictions on the smooth functions α and β, we have proved that a trans-Sasakian 3-manifold of type (α, β) admitting a ∗-Ricci soliton reduces to a β-Kenmotsu manifold and admitting a ∗-gradient Ricci soliton is either flat or ∗-Einstein or it becomes a β-Kenmotsu manifold. Also an illustrative example is presented to verify our results.

RIGIDITY OF GRADIENT SHRINKING AND EXPANDING RICCI SOLITONS

  • Yang, Fei;Zhang, Liangdi
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.3
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    • pp.817-824
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    • 2017
  • In this paper, we prove that a gradient shrinking Ricci soliton is rigid if the radial curvature vanishes and the second order divergence of Bach tensor is non-positive. Moreover, we show that a complete non-compact gradient expanding Ricci soliton is rigid if the radial curvature vanishes, the Ricci curvature is nonnegative and the second order divergence of Bach tensor is nonnegative.

ON GRADIENT RICCI SOLITONS AND YAMABE SOLITONS

  • Choi, Jin Hyuk;Kim, Byung Hak;Lee, Sang Deok
    • Journal of the Chungcheong Mathematical Society
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    • v.33 no.2
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    • pp.219-226
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    • 2020
  • In this paper, we consider gradient Ricci solitons and gradient Yamabe solitons in the warped product spaces. Also we study warped product space with harmonic curvature related to gradient Ricci solitons and gradient Yamabe solitons. Consequently some theorems are generalized and we derive differential equations for a warped product space to be a gradient Ricci soliton.

BETA-ALMOST RICCI SOLITONS ON ALMOST COKÄHLER MANIFOLDS

  • Kar, Debabrata;Majhi, Pradip
    • Korean Journal of Mathematics
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    • v.27 no.3
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    • pp.691-705
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    • 2019
  • In the present paper is to classify Beta-almost (${\beta}$-almost) Ricci solitons and ${\beta}$-almost gradient Ricci solitons on almost $CoK{\ddot{a}}hler$ manifolds with ${\xi}$ belongs to ($k,{\mu}$)-nullity distribution. In this paper, we prove that such manifolds with V is contact vector field and $Q{\phi}={\phi}Q$ is ${\eta}$-Einstein and it is steady when the potential vector field is pointwise collinear to the reeb vectoer field. Moreover, we prove that a ($k,{\mu}$)-almost $CoK{\ddot{a}}hler$ manifolds admitting ${\beta}$-almost gradient Ricci solitons is isometric to a sphere.

SOME RESULTS IN η-RICCI SOLITON AND GRADIENT ρ-EINSTEIN SOLITON IN A COMPLETE RIEMANNIAN MANIFOLD

  • Mondal, Chandan Kumar;Shaikh, Absos Ali
    • Communications of the Korean Mathematical Society
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    • v.34 no.4
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    • pp.1279-1287
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    • 2019
  • The main purpose of the paper is to prove that if a compact Riemannian manifold admits a gradient ${\rho}$-Einstein soliton such that the gradient Einstein potential is a non-trivial conformal vector field, then the manifold is isometric to the Euclidean sphere. We have showed that a Riemannian manifold satisfying gradient ${\rho}$-Einstein soliton with convex Einstein potential possesses non-negative scalar curvature. We have also deduced a sufficient condition for a Riemannian manifold to be compact which satisfies almost ${\eta}$-Ricci soliton.

SOME RESULTS ON CONCIRCULAR VECTOR FIELDS AND THEIR APPLICATIONS TO RICCI SOLITONS

  • CHEN, BANG-YEN
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.5
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    • pp.1535-1547
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    • 2015
  • A vector field on a Riemannian manifold (M, g) is called concircular if it satisfies ${\nabla}X^v={\mu}X$ for any vector X tangent to M, where ${\nabla}$ is the Levi-Civita connection and ${\mu}$ is a non-trivial function on M. A smooth vector field ${\xi}$ on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation: $$\frac{1}{2}L_{\xi}g+Ric={\lambda}g$$, where $L_{\xi}g$ is the Lie-derivative of the metric tensor g with respect to ${\xi}$, Ric is the Ricci tensor of (M, g) and ${\lambda}$ is a constant. A Ricci soliton (M, g, ${\xi}$, ${\lambda}$) on a Riemannian manifold (M, g) is said to have concircular potential field if its potential field is a concircular vector field. In the first part of this paper we determine Riemannian manifolds which admit a concircular vector field. In the second part we classify Ricci solitons with concircular potential field. In the last part we prove some important properties of Ricci solitons on submanifolds of a Riemannian manifold equipped with a concircular vector field.

h-almost Ricci Solitons on Generalized Sasakian-space-forms

  • Doddabhadrappla Gowda, Prakasha;Amruthalakshmi Malleshrao, Ravindranatha;Sudhakar Kumar, Chaubey;Pundikala, Veeresha;Young Jin, Suh
    • Kyungpook Mathematical Journal
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    • v.62 no.4
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    • pp.715-728
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    • 2022
  • The aim of this article is to study the h-almost Ricci solitons and h-almost gradient Ricci solitons on generalized Sasakian-space-forms. First, we consider h-almost Ricci soliton with the potential vector field V as a contact vector field on generalized Sasakian-space-form of dimension greater than three. Next, we study h-almost gradient Ricci solitons on a three-dimensional quasi-Sasakian generalized Sasakian-space-form. In both the cases, several interesting results are obtained.

CURVATURE ESTIMATES FOR GRADIENT EXPANDING RICCI SOLITONS

  • Zhang, Liangdi
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.3
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    • pp.537-557
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    • 2021
  • In this paper, we investigate the curvature behavior of complete noncompact gradient expanding Ricci solitons with nonnegative Ricci curvature. For such a soliton in dimension four, it is shown that the Riemann curvature tensor and its covariant derivatives are bounded. Moreover, the Ricci curvature is controlled by the scalar curvature. In higher dimensions, we prove that the Riemann curvature tensor grows at most polynomially in the distance function.

Some Triviality Characterizations on Gradient Almost Yamabe Solitons

  • Uday Chand De;Puja Sarkar;Mampi Howlader
    • Kyungpook Mathematical Journal
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    • v.63 no.4
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    • pp.639-645
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    • 2023
  • An almost Yamabe soliton is a generalization of the Yamabe soliton. In this article, we deduce some results regarding almost gradient Yamabe solitons. More specifically, we show that a compact almost gradient Yamabe soliton having non-negative Ricci curvature is trivial. Again, we prove that an almost gradient Yamabe soliton with a non-negative potential function and scalar curvature bound admitting an integral condition is trivial. Moreover, we give a characterization of a compact almost gradient Yamabe solitons.