$(h,\eta)$-Ricci-Bourguignon Soliton on the Poincare Disk $\mathbb{D}^2$

Abstract

We present a new concept of $(h,\eta)$-Ricci-Bourguignon Soliton on a Riemannian manifold $(M,g)$ defined by \begin{equation}\label{eq1} \mathrm{Ric}+\frac{h}{2}\,\mathcal{L}_X g=(\lambda+\rho\,\mathrm{Scal})\,g + \omega\,\eta\otimes\eta, \end{equation} where $\eta$ is a $1$-form, $h$ is a non-zero smooth function, and $\lambda$, $\rho$ and $\omega$ are real constants, denoted by \textbf{$(M,g,X,\lambda,\rho,\omega)$}. We then explicitly write this equation on the Poincar\'e disk $\mathbb{D}^2$ equipped with the hyperbolic metric in polar coordinates.

Keywords

• η)-Ricci-Bourguignon Soliton
• Hyperbolic Disk
• Beltrami-Laplacian.
• Poincaré Metric

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