发表机构
Université Cheikh Anta Diop de Dakar; Imam Khomeini International University（达喀尔易卜拉欣·塞赫·安塔·迪奥普大学; 伊玛目霍梅尼国际大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究双曲带及其单位切丛上的Killing场、Ricci孤立子与η-双曲Ricci孤立子，明确了相关几何性质、孤立子存在性及分类结果。
AI 中文摘要
我们研究双曲带$S_a = \{(x,y)\in\mathbb{R}^2:0<y<a\}$上的Killing向量场、Ricci孤立子与双曲Ricci孤立子，该双曲带配备度量$g_{S_a}=\frac{\pi^2}{a^2\sin^2(\pi y/a)}(dx^2+dy^2)$；同时研究其单位切丛$T^1S_a$上的对应对象，该单位切丛配备Sasaki度量$g^S$。在双曲带上，我们证明其Gauss曲率恒为-1，Killing场的李代数同构于$\mathfrak{sl}(2,\mathbb{R})$，所有Ricci孤立子均为平凡的（具有Killing势，$\lambda=-1$），且满足：若2-Killing场是共形的，或是具有$\lambda\neq0$的双曲Ricci孤立子的势，则该场必为Killing场。在单位切丛上，我们证明$(T^1S_a,g^S)$的Killing场李代数是4维的，同构于$\mathfrak{sl}(2,\mathbb{R})\oplus\mathbb{R}$；$g^S$不是Einstein度量，但具有常系数的η-Einstein性质，即$\operatorname{Ric}^S=-\frac{3}{2}\\,g^S+2\\,\eta\otimes\eta$；且$(T^1S_a,g^S)$不存在Ricci孤立子。对于双曲Ricci孤立子，我们证明：具有共形投影的Killing势、左不变势及$\theta$-不变势的双曲Ricci孤立子不存在，且一般的$\theta$-不变势的双曲Ricci孤立子问题可归约为双曲带上的方程组；任意势的分类问题仍未解决。最后，我们证明$g^S$的η-Ricci孤立子具有Killing势，且$\lambda=\frac{3}{2}$、$\mu=-2$；在相同势类中得到的η-双曲Ricci孤立子是平凡的，具有Killing势，且$\mu=-\frac{3}{2}$、$\nu=2$。
英文摘要
We study Killing vector fields, Ricci solitons and hyperbolic Ricci solitons on the hyperbolic strip $S_a=\{(x,y)\in\mathbb{R}^2:0<y<a\}$ endowed with the metric $g_{S_{a}}=\frac{π^2}{a^2\sin^2(πy/a)}(dx^2+dy^2)$, and on its unit tangent bundle $T^1S_a$ endowed with the Sasaki metric $g^S$. On the strip, we show that the Gauss curvature is constant equal to $-1$, that the Lie algebra of Killing fields is isomorphic to $\mathfrak{sl}(2,\mathbb{R})$, that every Ricci soliton is trivial (Killing potential, $λ=-1$), and that a $2$-Killing field which is conformal, or which is the potential of a hyperbolic Ricci soliton with $λ\neq0$, is a Killing field. On the unit tangent bundle, we prove that the Lie algebra of Killing fields of $(T^1S_a,g^S)$ is four-dimensional and isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathbb{R}$, that $g^S$ is not Einstein but is $η$-Einstein with constant coefficients, $\operatorname{Ric}^S=-\tfrac32\,g^S+2\,η\otimesη$, and that $(T^1S_a,g^S)$ admits no Ricci soliton. For hyperbolic Ricci solitons we prove non-existence for Killing, left-invariant, and $θ$-invariant potentials with conformal projection, and we reduce the general $θ$-invariant case to a system of equations on the strip; the classification for an arbitrary potential is left open. Finally, we show that the $η$-Ricci solitons of $g^S$ have a Killing potential and $λ=\tfrac32$, $μ=-2$, and that the $η$-hyperbolic Ricci solitons obtained in the same classes of potentials are the trivial ones, with a Killing potential, $μ=-\tfrac32$ and $ν=2$.
