We prove some results for the solitons of the Ricci–Bourguignon flow, generalizing the corresponding results for Ricci solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of Ricci–Bourguignon almost solitons and prove some results about them that generalize previous results for Ricci almost solitons. We also derive integral formulas for compact gradient Ricci–Bourguignon solitons and compact gradient Ricci–Bourguignon almost solitons. Finally, using the integral formula, we show that a compact gradient Ricci–Bourguignon almost soliton is isometric to a Euclidean sphere if it has constant scalar curvature or its associated vector field is conformal.

我们证明了 Ricci-Bourguignon 流的孤子的一些结果，概括了 Ricci 孤子的相应结果。从 Ricci 几乎孤子中获取动机，然后我们介绍了 Ricci-Bourguignon几乎孤子的概念，并证明了一些关于它们的结果，这些结果概括了 Ricci 几乎孤子的先前结果。我们还推导出了致密梯度 Ricci-Bourguignon 孤子和致密梯度 Ricci-Bourguignon 几乎孤子的积分公式。最后，使用积分公式，我们证明了紧凑梯度 Ricci-Bourguignon 几乎是孤子，如果它具有恒定的标量曲率或其相关的矢量场是共形的，那么它与欧几里得球体是等距的。