Some basic properties of Ricci almost solitons

Ricci solitons are stationary solutions of a famous PDE for Riemannian metrics, known under the name of Ricci flow equation. An almost Ricci soliton is a remarkable generalization of Ricci solitons by allowing the soliton constant in Ricci flow equation to be a smooth function. In the present paper, we focuss our study on the most important class of almost Ricci solitons, namely gradient Ricci almost solitons $(M^n,g,\nabla \sigma ,f)$ with potential function σ and associated function f, abbreviated as GRRAS $(M^n,g,\nabla \sigma ,f)$. On a nontrivial GRRAS $(M^n,g,\nabla \sigma ,f)$, these two functions σ and f together with scalar curvature τ play a significant role. Among the basic properties of a connected GRRAS $(M^n,g,\nabla \sigma ,f)$, it has been observed that there exists a smooth function δ called the connector of the GRRAS $(M^n,g,\nabla \sigma ,f)$ as it connects the gradients of the potential function σ and the associated function f, respectively. In our first result it is shown that a nontrivial GRRAS $(M^n,g,\nabla \sigma ,f)$ with connector δ gives a generalized soliton, thus establishing an unexpected duality. In our second result, we show that a compact and connected nontrivial GRRAS $(M^n,g,\nabla \sigma ,f)$ with connector $\delta =-c$, for a positive constant c, and a suitable lower bound on the integral of the Ricci curvature $Ric(\nabla \sigma ,\nabla \sigma )$ is isometric to the n-sphere $S^n\left(c\right)$ and the converse too is shown to hold. In the third result it is established that a complete and simply connected nontrivial GRRAS $(M^n,g,\nabla \sigma ,f)$ of positive scalar curvature, with a suitable lower bound on $Ric(\nabla \sigma ,\nabla \sigma )$ and the vector ∇σ being eigenvector of the Hessian operator $H_{\sigma }$ with an appropriate eigenvalue, gives a characterization of $S^n\left(c\right)$. In our final result, we consider a compact and connected nontrivial GRRAS $(M^n,g,\nabla \sigma ,f)$ of positive scalar curvature and ask the associated function f to satisfy a Poison equation to get yet other characterization of $S^n\left(c\right)$.