• Korean Journal of Mathematics
• /
• v.28 no.4
• /
• pp.803-817
• /
• 2020

The object of the present paper is to study 𝜂-Ricci soliton on Kenmotsu manifold with respect to general connection.

• Communications of the Korean Mathematical Society
• /
• v.38 no.3
• /
• pp.865-880
• /
• 2023

The main intention of the current paper is to characterize certain properties of *-conformal Ricci solitons on non-coKähler (𝜅, 𝜇)-almost coKähler manifolds. At first, we find that there does not exist *-conformal Ricci soliton if the potential vector field is the Reeb vector field θ. We also prove that the non-coKähler (𝜅, 𝜇)-almost coKähler manifolds admit *-conformal Ricci solitons if the potential vector field is the infinitesimal contact transformation. It is also studied that there does not exist *-conformal gradient Ricci solitons on the said manifolds. An example has been constructed to verify the obtained results.

• Communications of the Korean Mathematical Society
• /
• v.37 no.2
• /
• pp.585-594
• /
• 2022

In this article we classify four dimensional gradient Ricci solitons (M, g, f) with half harmonic Weyl curvature and at most two distinct Ricci-eigenvalues at each point. Indeed, we showed that, in a neighborhood V of each point in some open dense subset of M, (V, g) is isometric to one of the following: (i) an Einstein manifold. (ii) a domain in the Riemannian product (ℝ², g₀) × (N, ${\tilde{g}}$), where g₀ is the flat metric on ℝ² and (N, ${\tilde{g}}$) is a two dimensional Riemannian manifold of constant curvature λ ≠ 0. (iii) a domain in ℝ × W with the warped product metric $ds^2+h(s)^2{\tilde{g}}$, where ${\tilde{g}}$ is a constant curved metric on a three dimensional manifold W.

• Bulletin of the Korean Mathematical Society
• /
• v.60 no.6
• /
• pp.1673-1685
• /
• 2023

Inspired by Hongjie Dong and Qi S. Zhang's article [3], we find that the analyticity in time for a smooth solution of the heat equation with exponential quadratic growth in the space variable can be extended to any complete noncompact Riemannian manifolds with Bakry-Émery Ricci curvature bounded below and the potential function being of at most quadratic growth. Therefore, our result holds on all gradient Ricci solitons. As a corollary, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with the similar growth condition. In addition, we also consider the solution in certain L^p spaces with p ∈ [2, +∞) and prove its analyticity with respect to time.

• Honam Mathematical Journal
• /
• v.41 no.3
• /
• pp.539-558
• /
• 2019

In this paper, we study ${\eta}$-Ricci solitons on ${\epsilon}$-LP-Sasakian manifolds with a quarter-symmetric metric connection satisfying certain curvature conditions. In particular, we have discussed that the Ricci soliton on ${\epsilon}$-LP-Sasakian manifolds with a quarter-symmetric metric connection satisfying certain curvature conditions is expanding or steady according to the vector field ${\xi}$ being timelike or spacelike. Moreover, we construct 3-dimensional examples of an ${\epsilon}$-LP-Sasakian manifold with a quarter-symmetric metric connection to verify some results of the paper.

• Journal of the Korean Mathematical Society
• /
• v.55 no.6
• /
• pp.1285-1303
• /
• 2018

This note is motivated by gradient estimates of Li-Yau, Hamilton, and Souplet-Zhang for heat equations. In this paper, our aim is to investigate Yamabe equations and a non linear heat equation arising from gradient Ricci soliton. We will apply Bochner technique and maximal principle to derive gradient estimates of the general non-linear heat equation on Riemannian manifolds. As their consequence, we give several applications to study heat equation and Yamabe equation such as Harnack type inequalities, gradient estimates, Liouville type results.

• Communications of the Korean Mathematical Society
• /
• v.36 no.2
• /
• pp.361-375
• /
• 2021

The object of the present paper is to deduce some important results on generic submanifolds and also generic product of LP-Sasakian manifolds with concurrent vector fields. Also, we provide a necessary and sufficient condition for which the invariant distribution D and anti-invariant distribution D^⊥ of M are Einstein. Also, we deduce an interesting necessary and sufficient condition for submanifolds of LP-Sasakian manifolds to be totally umbilical submanifolds. Especially we deal with the generic submanifolds admitting a Ricci soliton in LP-Sasakian manifolds endowed with concurrent vector fields.