• Korean Journal of Mathematics
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• v.28 no.4
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• pp.739-751
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• 2020

The object of the offering exposition is to study almost Ricci soliton and gradient almost Ricci soliton in 3-dimensional para-Sasakian manifolds. At first, it is shown that if (g, V, λ) be an almost Ricci soliton on a 3-dimensional para-Sasakian manifold M, then it reduces to a Ricci soliton and the soliton is expanding for λ=-2. Besides these, in this section, we prove that if V is pointwise collinear with ξ, then V is a constant multiple of ξ and the manifold is of constant sectional curvature -1. Moreover, it is proved that if a 3-dimensional para-Sasakian manifold admits gradient almost Ricci soliton under certain conditions then either the manifold is of constant sectional curvature -1 or it reduces to a gradient Ricci soliton. Finally, we consider an example to justify some results of our paper.

• 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.

• 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.

• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.1101-1114
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• 2016

Let ($M^{2n+1}$, ${\phi}$, ${\xi}$, ${\eta}$, g) be a (k, ${\mu}$)'-almost Kenmotsu manifold with k < -1 which admits a gradient Ricci almost soliton (g, f, ${\lambda}$), where ${\lambda}$ is the soliton function and f is the potential function. In this paper, it is proved that ${\lambda}$ is a constant and this implies that $M^{2n+1}$ is locally isometric to a rigid gradient Ricci soliton ${\mathbb{H}}^{n+1}(-4){\times}{\mathbb{R}}^n$, and the soliton is expanding with ${\lambda}=-4n$. Moreover, if a three dimensional Kenmotsu manifold admits a gradient Ricci almost soliton, then either it is of constant sectional curvature -1 or the potential vector field is pointwise colinear with the Reeb vector field.