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 flflow 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 (Mn,g,∇σ,f)with potential function σand associated function f, abbreviated as GRRAS (Mn,g,∇σ,f). On a nontrivial GRRAS (Mn,g,∇σ,f), these two functions σand ftogether with scalar curvature τplay a significant role. Among the basic properties of a connected GRRAS (Mn,g,∇σ,f), it has been observed that there exists a smooth function δcalled the connector of the GRRAS (Mn,g,∇σ,f)as it connects the gradients of the potential function σand the associated function f, respectively. In our fifirst result it is shown that a nontrivial GRRAS (Mn,g,∇σ,f)with connector δgives a generalized soliton, thus establishing an unexpected duality. In our second result, we show that a compact and connected nontrivial GRRAS (Mn,g,∇σ,f)with connector δ=−c, for a positive constant c, and a suitable lower bound on the integral of the Ricci curvature Ric(∇σ,∇σ)is isometric to the n-sphere Sn(c)and the converse too is shown to hold. In the third result it is established that a complete and simply connected nontrivial GRRAS (Mn,g,∇σ,f)of positive scalar curvature, with a suitable lower bound on Ric(∇σ,∇σ)and the vector ∇σbeing eigenvector of the Hessian operator Hσwith an appropriate eigenvalue, gives a characterization of Sn(c). In our fifinal result, we consider a compact and connected nontrivial GRRAS (Mn,g,∇σ,f)of positive scalar curvature and ask the associated function fto satisfy a Poison equation to get yet other characterization of Sn(c).
Ricci solitons are stationary solutions of a famous PDE for Riemannian metrics, known under the name of Ricci flflow equation. An almost Ricci soliton is a remarkable generalization of Ricci…
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