• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.213-228
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• 2022

In the present paper, we aim to study Yamabe soliton and Riemann soliton on Lorentzian para-Sasakian manifold. First, we proved, if the scalar curvature of an 𝜂-Einstein Lorentzian para-Sasakian manifold M is constant, then either 𝜏 = n(n-1) or, 𝜏 = n-1. Also we constructed an example to justify this. Next, it is proved that, if a three dimensional Lorentzian para-Sasakian manifold admits a Yamabe soliton for V is an infinitesimal contact transformation and tr 𝜑 is constant, then the soliton is expanding. Also we proved that, suppose a 3-dimensional Lorentzian para-Sasakian manifold admits a Yamabe soliton, if tr 𝜑 is constant and scalar curvature 𝜏 is harmonic (i.e., ∆𝜏 = 0), then the soliton constant λ is always greater than zero with either 𝜏 = 2, or 𝜏 = 6, or λ = 6. Finally, we proved that, if an 𝜂-Einstein Lorentzian para-Sasakian manifold M represents a Riemann soliton for the potential vector field V has constant divergence then either, M is of constant curvature 1 or, V is a strict infinitesimal contact transformation.

• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.881-892
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• 2023

In this paper, we study almost cosymplectic manifolds with nullity distributions admitting Riemann solitons and gradient almost Riemann solitons. First, we consider Riemann soliton on (κ, µ)-almost cosymplectic manifold M with κ < 0 and we show that the soliton is expanding with ${\lambda}{\frac{\kappa}{2n-1}}(4n - 1)$ and M is locally isometric to the Lie group G_ρ. Finally, we prove the non-existence of gradient almost Riemann soliton on a (κ, µ)-almost cosymplectic manifold of dimension greater than 3 with κ < 0.

• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.1215-1231
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• 2023

The present article contains the study of D-homothetically deformed f-Kenmotsu manifolds. Some fundamental results on the deformed spaces have been deduced. Some basic properties of the Riemannian metric as an inner product on both the original and deformed spaces have been established. Finally, applying the obtained results, soliton functions, Ricci curvatures and scalar curvatures of almost Riemann solitons with several kinds of potential vector fields on the deformed spaces have been characterized.

• 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.