Soumendu Roy, Santu Dey, Arindam Bhattacharyya
The object of the present paper is to study some properties of para-K$ä$hler manifold whose metric is conformal Einstein soliton. We have studied some certain curvature properties of para-K$ä$hler manifold admitting conformal Einstein soliton.
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We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log terminal singularities.
Let Z be the real part of a K{ä}hler Hamiltonian manifold M. The O'Shea-Sjamaar's Theorem tells us that the moment polytope Delta(Z) corresponds to the anti-invariant part of the Kirwan polytope Delta(M). The purpose of the present paper is to explain how to parameterize the equations of the facets of Delta(Z) in terms of real Ressayre's pairs of Z.
In this paper, we initiate the study of conformal $η$-Ricci soliton and almost conformal $η$-Ricci soliton within the framework of para-Sasakian manifold. We prove that if para-Sasakian metric admits conformal $η$-Ricci soliton, then the manifold is $η$-Einstein and either the soliton vector field $V$ is Killing or it leaves $ϕ$ invariant. Here, we have shown the characteristics of the soliton vector field $V$ and scalar curvature when the manifold admitting conformal $η$-Ricci soliton and vector field is pointwise collinear with the characteristic vector field $ξ$. Next, we show that a para-S…
Gradient almost para-Ricci-like solitons on para-Sasaki-like Riemannian $Π$-manifolds are studied. It is proved that these objects have constant soliton coefficients. For the soliton under study is shown that the corresponding scalar curvatures of the considered both metrics are equal and constant and its Ricci tensor is a constant multiple of the vertical component. Explicit example of a 3-dimensional para-Sasaki-like Riemannian $Π$-manifold is provided in support of the proved assertions.
In this paper we study a special type of metric called *-Ricci soliton on para-Sasakian manifold. We prove that if the para-Sasakian metric is a *-Ricci soliton on a manifold M, then M is either D-homothetic to an Einstein manifold, or the Ricci tensor of M with respect to the canonical paracontact connection vanishes.
Para-Ricci-like solitons with arbitrary potential on para-Sasaki-like Riemannian $Π$-manifolds are introduced and studied. For the studied soliton, it is proved that its Ricci tensor is a constant multiple of the vertical component of both metrics. Thus, the corresponding scalar curvatures of both considered metrics are equal and constant. An explicit example of the Lie group as the manifold under study is presented.
H-almost Ricci-Yamabe solitons in paracontact geometry
本文在仿切触几何框架下分类了 h-几乎 Ricci-Yamabe 孤立子，系统刻画了 para-Kenmotsu、para-Sasakian 和 para-cosymplectic 流形上的相关结构，并给出两个实例加以说明。
Object of study are para-Ricci-like solitons on para-Sasaki-like almost paracontact almost paracomplex Riemannian manifolds, briefly, Riemannian $Π$-manifolds. Different cases when the potential of the soliton is the Reeb vector field or pointwise collinear to it are considered. Some additional geometric properties of the constructed objects are proven. Results for a parallel symmetric second-order covariant tensor on the considered manifolds are obtained. Explicit example of dimension 5 in support of the given assertions is provided.
Geometry of δ-almost gradient Yamabe solitons on pseudo-Riemannian manifolds
本文研究伪黎曼流形中仿切触度量流形上的δ-几乎Yamabe孤立子，证明势向量场的Killing性、与ξ的平行关系及数量曲率条件，并构造了两个K-仿切触度量流形实例。
$η$-Ricci solitons in $(\varepsilon)$-almost paracontact metric manifolds
该文针对$(\u03b5)$-殆仿切触度量流形，研究了两类势向量场情形下的$\u03b7$-Ricci孤立子，分析了相关特殊流形的孤立子容许性，并结合平行对称(0,2)张量场得到了若干研究结果。
The purpose of the paper is to study Yamabe solitons on three-dimensional para-Sasakian, paracosymplectic and para-Kenmotsu manifolds. Mainly, we proved that *If the semi-Riemannian metric of a three-dimensional para-Sasakian manifold is a Yamabe soliton, then it is of constant scalar curvature, and the flow vector field V is Killing. In the next step, we proved that either manifold has constant curvature -1 and reduces to an Einstein manifold, or V is an infinitesimal automorphism of the paracontact metric structure on the manifold. *If the semi-Riemannian metric of a three-dimensional paraco…
Remarks on almost $η$-Ricci solitons in $(\varepsilon)$-para Sasakian manifolds
本文研究满足特定曲率条件的$(\u03b5)$-para Sasakian流形中几乎$\u03b7$-Ricci孤立子，给出梯度情形下Ricci曲率范数估计与标量曲率表达式，并证明Codazzi型Ricci算子下类空/类时流形对应膨胀/收缩孤立子。
The Eisenhart problem of finding parallel tensors treated already in the framework of quasi-constant curvature manifolds in \cite{x:j} is reconsidered for the symmetric case and the result is interpreted in terms of Ricci solitons. If the generator of the manifold provides a Ricci soliton then this is i) expanding on para-Sasakian spaces with constant scalar curvature and vanishing $D$-concircular tensor field and ii) shrinking on a class of orientable quasi-umbilical hypersurfaces of a real projective space=elliptic space form.
We show that a complete gradient Ricci soliton $(M^n,\,g)$ with constant scalar curvature and a non-parallel closed conformal vector field is isometric to either the Euclidean space, or an Euclidean sphere, or negatively Einstein warped product of the real line with a complete non-positively Einstein manifold. Moreover, we show that a Kähler gradient Ricci soliton of real dimension $\ge 4$, with a non-parallel closed conformal real vector field is Ricci-flat (Calabi-Yau) and is flat in dimension 4. Finally, we show that a Ricci soliton whose associated 1-form is harmonic, has constant scalar c…
$η$-Ricci solitons on para-Kenmotsu manifolds
本文在仿切触几何背景下研究满足特定曲率条件的para-Kenmotsu流形上的$η$-Ricci孤立子，证明其存在蕴含流形拟爱因斯坦，满足特定曲率条件时为爱因斯坦流形，并给出其存在的充分条件。
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