Some characterizations on gradient almost $\eta$-Ricci-Bourguignon solitons
Abstract
We characterize a Riemannian manifold with gradient almost $\eta$-Ricci Bourguignon solitons structures. We show that a gradient almost $\eta$-Ricci-Bourguignon soliton is gradient $(-\frac{1}{\omega u})$-almost traceless Ricci soliton with the potential function $k$. Moreover, we investigate that a gradient $(-\frac{1}{\omega u})$-almost traceless Ricci soliton is isometric to a standard unit sphere $\mathbb{S}^{n}$, hyperbolic space $\mathbb{H}^{n}$ and Euclidean space $\mathbb{R}^{n}$ with constant scalar curvature or its associated vector fields is conformal. Finally, we deduce some properties of integral formulas for the gradient compact case.
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