Some Characteristics of Almost $\omega$-Bach Solitons

In this article, we introduce $\omega$-Bach tensor corresponding to one form $\omega$ and correspondingly introduce almost $\omega$-Bach solitons, thereby generalizing the existing notion of Bach tensor and almost Bach solitons. We characterize almost $\omega$-Bach solitons, when the potential vector field of the soliton generates an infinitesimal harmonic transformation or is an affine conformal vector field, or is a projective vector field or is a Killing vector field, when the $\omega$-Bach tensor is divergence free, or is a harmonic $1$ form or is a Killing $1$-form. We generalize some of the results obtained by P. T. Ho and A. Ghosh. One of the main results of this paper is that we explicitly find some of the gradient almost $\omega$-Bach solitons on the product manifolds ${\mathbb S}^2\times{\mathbb H}^2$, $\mathbb{R}^2\times{\mathbb H}^2$ and $\mathbb{R}^2\times{\mathbb S}^2$. Our gradient almost $\omega$-Bach solitons generalize the almost Bach solitons on $\mathbb{R}^2\times{\mathbb H}^2$ and $\mathbb{R}^2\times{\mathbb S}^2$ found by P. T. Ho. Moreover, finding of our gradient almost $\omega$-Bach solitons on ${\mathbb S}^2\times{\mathbb H}^2$ is a novel one and complements to the existing almost Bach solitons described by P. T. Ho.