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GRADIENT YAMABE SOLITONS WITH CONFORMAL VECTOR FIELD
Fasihi-Ramandi, Ghodratallah,Ghahremani-Gol, Hajar Korean Mathematical Society 2021 대한수학회논문집 Vol.36 No.1
The purpose of this paper is to investigate the geometry of complete gradient Yamabe soliton (Mn, g, f, λ) with constant scalar curvature admitting a non-homothetic conformal vector field V leaving the potential vector field invariant. We show that in such manifolds the potential function f is constant and the scalar curvature of g is determined by its soliton scalar. Considering the locally conformally flat case and conformal vector field V, without constant scalar curvature assumption, we show that g has constant curvature and determines the potential function f explicitly.
Harnack Estimate for Positive Solutions to a Nonlinear Equation Under Geometric Flow
Fasihi-Ramandi, Ghodratallah,Azami, Shahroud Department of Mathematics 2021 Kyungpook mathematical journal Vol.61 No.3
In the present paper, we obtain gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds $$\frac{{\partial}u}{{\partial}t}={\Delta}u+a(x,t)u^p+b(x,t)u^q$$ where, 0 < p, q < 1 are real constants and a(x, t) and b(x, t) are functions which are C<sup>2</sup> in the x-variable and C<sup>1</sup> in the t-variable. We shall get an interesting Harnack inequality as an application.
THE SET OF ZOLL METRICS IS NOT PRESERVED BY SOME GEOMETRIC FLOWS
Azami, Shahroud,Fasihi-Ramandi, Ghodratallah Korean Mathematical Society 2019 대한수학회논문집 Vol.34 No.3
The geodesics on the round 2-sphere $S^2$ are all simple closed curves of equal length. In 1903 Otto Zoll introduced other Riemannian surfaces with the same property. After that, his name is attached to the Riemannian manifolds whose geodesics are all simple closed curves of the same length. The question that "whether or not the set of Zoll metrics on 2-sphere $S^2$ is connected?" is still an outstanding open problem in the theory of Zoll manifolds. In the present paper, continuing the work of D. Jane for the case of the Ricci flow, we show that a naive application of some famous geometric flows does not work to answer this problem. In fact, we identify an attribute of Zoll manifolds and prove that along the geometric flows this quantity no longer reflects a Zoll metric. At the end, we will establish an alternative proof of this fact.
RICCI 𝜌-SOLITONS ON 3-DIMENSIONAL 𝜂-EINSTEIN ALMOST KENMOTSU MANIFOLDS
Azami, Shahroud,Fasihi-Ramandi, Ghodratallah Korean Mathematical Society 2020 대한수학회논문집 Vol.35 No.2
The notion of quasi-Einstein metric in theoretical physics and in relation with string theory is equivalent to the notion of Ricci soliton in differential geometry. Quasi-Einstein metrics or Ricci solitons serve also as solution to Ricci flow equation, which is an evolution equation for Riemannian metrics on a Riemannian manifold. Quasi-Einstein metrics are subject of great interest in both mathematics and theoretical physics. In this paper the notion of Ricci 𝜌-soliton as a generalization of Ricci soliton is defined. We are motivated by the Ricci-Bourguignon flow to define this concept. We show that if a 3-dimensional almost Kenmotsu Einstein manifold M is a 𝜌-soliton, then M is a Kenmotsu manifold of constant sectional curvature -1 and the 𝜌-soliton is expanding with λ = 2.
Generalized hyperbolic geometric flow
Shahroud Azami,Ghodratallah Fasihi-Ramandi,Vahid Pirhadi 대한수학회 2023 대한수학회논문집 Vol.38 No.2
In the present paper, we consider a kind of generalized hyperbolic geometric flow which has a gradient form. Firstly, we establish the existence and uniqueness for the solution of this flow on an $n$-dimensional closed Riemannian manifold. Then, we give the evolution of some geometric structures of the manifold along this flow.
Siddiqi, Mohammed Danish,Chaubey, Sudhakar Kumar,Ramandi, Ghodratallah Fasihi Department of Mathematics 2021 Kyungpook mathematical journal Vol.61 No.3
This paper examines the behavior of a 3-dimensional trans-Sasakian manifold equipped with a gradient generalized quasi-Yamabe soliton. In particular, It is shown that α-Sasakian, β-Kenmotsu and cosymplectic manifolds satisfy the gradient generalized quasi-Yamabe soliton equation. Furthermore, in the particular case when the potential vector field ζ of the quasi-Yamabe soliton is of gradient type ζ = grad(ψ), we derive a Poisson's equation from the quasi-Yamabe soliton equation. Also, we study harmonic aspects of quasi-Yamabe solitons on 3-dimensional trans-Sasakian manifolds sharing a harmonic potential function ψ. Finally, we observe that 3-dimensional compact trans-Sasakian manifold admits the gradient generalized almost quasi-Yamabe soliton with Hodge-de Rham potential ψ. This research ends with few examples of quasi-Yamabe solitons on 3-dimensional trans-Sasakian manifolds.