Conformal \(\eta\)-Ricci-Yamabe solitons in the framework of Riemannian manifolds (Q6619041)

The authors introduce certain generalized conformal Ricci-Yamabe solitons, named conformal \(\eta\)-Ricci-Yamabe solitons (CERYS), in the context of Riemannian manifolds \((M^n,g)\). A Riemannian manifolds is said to satisfy a CERYS of type \((p,q)\) if\N\begin{align*}\N\mathcal{L}_F g + 2pS +\left\{2\sigma - qr -\left(\psi+\frac{2}{n}\right)\right\}g + 2\eta\otimes \eta = 0,\N\end{align*}\Nwhere \(\mathcal{L}\) is the Lie derivative operator along the smooth function \(F\) on \(M^n\), \(\sigma, \rho \in \mathbb{R}\). If \(F\) is the gradient of a smooth function \(f\) on \(M^n\), the the corresponding gradient CERYS can be obtained with the Hessian of \(f\). A CERYS is shrinking, steady or expanding if \(\sigma < 0, \sigma = 0\) or \(\sigma > 0\), respectively. Clearly, CERYS include conformal Ricci solitons, conformal Yamabe solitons, conformal Einstein solitons, etc. The authors further study CERYS and its corresponding gradient forms on \(3\)-dimensional manifolds, highlighting their properties and then constructing a non-trivial example.\N\NFor the entire collection see [Zbl 1537.53001].