$\boldsymbol{\text{H}^2\times\boldsymbol{\text{R}}}$上的Ricci孤立子分类
Ricci Soliton Classification on $\mathbb H^2\times\mathbb R$
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中文总结 AI 辅助
本文对配备标准度量的黎曼流形$\boldsymbol{\text{H}^2\times\boldsymbol{\text{R}}}$上的Ricci孤立子进行分类，明确其孤立子向量场的结构、类型及性质，得到梯度孤立子的子族特征。
中文摘要 AI 辅助
我们研究配备标准度量的黎曼流形$\boldsymbol{\text{H}^2\times\boldsymbol{\text{R}}}$上的Ricci孤立子，得到了孤立子向量场的完整分类：它们构成一个四维仿射空间，即Killing代数$\boldsymbol{\text{isom}}(\boldsymbol{\text{H}^2\times\boldsymbol{\text{R}}})$的平移。所有对应的孤立子都是膨胀型的。此外，梯度孤立子得到了完全刻画，且被证明是孤立子场完整族的一个单参数子族。作为附带结果，每个孤立子向量场都是仿射的，保持Levi-Civita联络、曲率张量和Ricci张量。
英文摘要
We study Ricci solitons on the Riemannian manifold $\mathbb H^2\times\mathbb R$ equipped with the standard metric. A complete classification of soliton vector fields is obtained: they form a four-dimensional affine space, namely a translate of the Killing algebra $\mathfrak{isom}(\mathbb H^2\times\mathbb R)$. All corresponding solitons are expanding. In addition, gradient solitons are fully characterized and shown to form a one-parameter subfamily of the complete family of soliton fields. As a byproduct, every soliton vector field turns out to be affine, preserving the Levi-Civita connection, the curvature tensor, and the Ricci tensor.
