AI 中文总结
研究具有常长度势场的有限体积非紧Ricci孤子，在特定假设下证明其平凡性，作为应用得到相关Ricci孤子的刚性和不存在性结果，在三维中有进一步推导及与相关分类比较，还给出例子和反例说明假设必要性。
AI 中文摘要
我们研究势向量场具有常长度的有限体积非紧Ricci孤子。在标量曲率沿势场的积分曲线为常数以及一个自然散度项在单位切丛上可积的假设下，我们证明这样的Ricci孤子必然是平凡的。作为应用，我们得到了势场为几乎接触度量和几乎\(\alpha\)-余辛流形的Reeb向量场的Ricci孤子的刚性和不存在性结果。在三维中，我们推导了几乎\(\alpha\)-余辛和接触度量流形的结果，并将我们的结果与Li和Liu给出的齐次几乎\(\alpha\)-余辛Ricci孤子的分类进行比较。包含了几个例子和反例来说明有限体积和符号假设的必要性。
英文摘要
We study non-compact Ricci solitons of finite volume whose potential vector field has constant length. Under the assumptions that the scalar curvature is constant along the integral curves of the potential field and that a natural divergence term is integrable on the unit tangent bundle, we prove that such Ricci solitons are necessarily trivial. As applications, we obtain rigidity and non-existence results for Ricci solitons whose potential field is the Reeb vector field of almost contact metric and almost $α$-cosymplectic manifolds. In dimension three, we derive consequences for almost $α$-cosymplectic and contact metric manifolds, and we compare our results with the classification of homogeneous almost $α$-cosymplectic Ricci solitons due to Li and Liu. Several examples and non-examples are included to illustrate the necessity of the finite-volume and sign assumptions.
