切触度量流形上的几乎Ricci-Yamabe孤子
Almost Ricci-Yamabe Soliton on Contact Metric Manifolds
摘要：
我们考虑在某些接触度量流形的背景下几乎Ricci-Yamabe孤子。首先，证明了若度量$g$在完备的切触度量流形上存在几乎$(α,β)$ -Ricci-Yamabe孤子，其中$α\neq 0$和势向量场与Reeb向量场$ξ$共线，且Reeb向量场$ξ$是Ricci算子的特征向量，则该流形是紧Einstein Sasaki流形，势向量场是Reeb向量场$ξ$的常倍数.其次，如果完备的$K$ -contact流形中存在梯度$α\neq 0$孤子，则它是紧的Sasakian，且等距于单位球$S^{2n+1}$ .最后，在非Sasakian的$(k,μ)$ -接触度量流形中，研究了具有$α\neq 0$的梯度几乎Ricci-Yamabe孤子，发现$M^3$是平坦的，对于$n>1$，$M$与$E^{n+1}\times S^n(4)$局部等距，孤子矢量场与欧氏因子$E^{n+1}$相切.最后给出了一个例子来支持所得到的结果。
英文摘要：
We consider almost Ricci-Yamabe soliton in the context of certain contact metric manifolds. Firstly, we prove that if the metric $g$ admits an almost $(α,β)$-Ricci-Yamabe soliton with $α\neq 0$ and potential vector field collinear with the Reeb vector field $ξ$ on a complete contact metric manifold with the Reeb vector field $ξ$ as an eigenvector of the Ricci operator, then the manifold is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field $ξ$. Next, if complete $K$-contact manifold admits gradient Ricci-Yamabe soliton with $α\neq 0$, then it is compact Sasakian and isometric to unit sphere $S^{2n+1}$. Finally, gradient almost Ricci-Yamabe soliton with $α\neq 0$ in non-Sasakian $(k,μ)$-contact metric manifold is assumed and found that $M^3$ is flat and for $n>1$, $M$ is locally isometric to $E^{n+1}\times S^n(4)$ and the soliton vector field is tangential to the Euclidean factor $E^{n+1}$. An illustrative example is given to support the obtained result.
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