AI 中文总结
研究三维接触度量流形上几乎 Ricci - Bourguignon 孤子，利用局部正交\(\varphi\)-基推导方程分量形式，探讨势向量场与 Reeb 向量场共线或正交情况，得出共线时特定条件下势场为零、正交时度量相关结论等贡献。
AI 中文摘要
我们研究三维接触度量流形上的几乎 Ricci - Bourguignon 孤子。在自然曲率假设下，表明允许孤子函数变化所引入的额外自由度受到接触几何的严格约束。利用非 Sasakian 区域上的局部正交\(\varphi\)-基，推导了几乎 Ricci - Bourguignon 孤子方程的完整分量形式。作为应用，考虑了势向量场与 Reeb 向量场逐点共线或正交的情况。对于满足\(Q\xi=\sigma\xi\)的接触度量三维流形，证明了当\(\xi(\sigma)=0\)时，共线势场在非 Sasakian 区域必定为零。在正交情况下，当\(\sigma\)为常数且流形是非 Sasakian 时，几乎孤子函数被迫为常数，孤子简化为 Ricci - Bourguignon 孤子，实际上度量是 Einstein 的，若正交势场不全为零，则度量是平坦的。
英文摘要
We investigate almost Ricci--Bourguignon solitons on three-dimensional contact metric manifolds. Under natural curvature assumptions, we show that the additional freedom introduced by allowing the soliton function to vary is rigidly constrained by the contact geometry. Using a local orthonormal \(φ\)-basis on the non-Sasakian region, we derive the full component form of the almost Ricci--Bourguignon soliton equation. As applications, we consider the cases where the potential vector field is pointwise collinear with, or orthogonal to, the Reeb vector field. For contact metric three-manifolds satisfying \(Qξ=σξ\), we prove that a collinear potential field must vanish on the non-Sasakian region whenever \(ξ(σ)=0\). In the orthogonal case, when \(σ\) is constant and the manifold is non-Sasakian, the almost soliton function is forced to be constant; hence the soliton reduces to a Ricci--Bourguignon soliton. In fact, the metric is Einstein, and if the orthogonal potential field is not identically zero, then the metric is flat.
