发表机构
Université Cheikh Anta Diop de Dakar（达喀尔易卜拉欣·塞克·安塔·迪奥普大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究秩一各向异性曲率黎曼流形上，Ricci孤立子蕴含几乎θ-Yamabe孤立子，并证明向量场对称性条件等价于由算子D_X=X+6φ+2a控制的标量微分问题。
AI 中文摘要
我们研究了具有秩一各向异性曲率的黎曼流形上李导数对曲率张量的迭代作用，其黎曼张量通过Kulkarni-Nomizu积表示为$R = \lambda (\xi^\flat\otimes\xi^\flat)\owedge g$。首先，我们考察了此类流形允许Ricci孤立子结构的条件，并证明了该性质蕴含几乎$\theta$-Yamabe孤立子结构。此外，我们证明：若相关的势向量场$X$是Ricci张量的固定阶$k$的对称性（即$\mathcal{L}_X^k \operatorname{Ric} = 0$），则几何问题归结为沿流求解一个$k+1$阶偏微分方程。最后，在假设$X$是共形向量场（$\mathcal{L}_X g = 2\varphi g$）且其无穷小流保持线分布$\mathcal{D}=\operatorname{Span}\{\xi\}$（其中$[X,\xi]=a\xi$，$a\in\mathbb{R}$）的条件下，我们证明了几个关键几何问题（如建立关系$\mathcal{L}_X^k R = R$、确定$X$成为李曲率对称性的最小阶$k$、或满足$\mathcal{L}_X^{k+1}R = f \mathcal{L}_X^k R$其中$f$为连续函数）等价于由算子$D_X = X + 6\varphi + 2a$支配的标量微分问题。
英文摘要
We study the iterated action of the Lie derivative on the curvature tensor of a Riemannian manifold with rank-one anisotropic curvature, whose Riemann tensor is expressed via the Kulkarni-Nomizu product as $R = λ(ξ^\flat\otimesξ^\flat)\owedge g$. First, we examine the conditions under which such a manifold admits a Ricci soliton structure and demonstrate that this property implies the almost $θ$-Yamabe soliton structure. Furthermore, we show that if the associated potential vector field $X$ is a symmetry of the Ricci tensor of a fixed order $k$ (i.e., $\mathcal{L}_X^k \operatorname{Ric} = 0$), the geometric problem reduces to solving a partial differential equation of order $k+1$ along the flow. Finally, under the assumption that $X$ is a conformal vector field ($\mathcal{L}_X g = 2φg$) whose infinitesimal flow preserves the line distribution $\mathcal{D}=\operatorname{Span}\{ξ\}$ (with $[X,ξ]=aξ$ for $a\in\mathbb{R}$), we prove that several key geometric problems (such as establishing the relation $\mathcal{L}_X^k R = R$, determining the minimal order $k$ for $X$ to be a Lie curvature symmetry, or satisfying $\mathcal{L}_X^{k+1}R = f \mathcal{L}_X^k R$ for a continuous function $f$) are equivalent to a scalar differential problem governed by the operator $D_X = X + 6φ+ 2a$.
Comments30 pages
