Characterization of Almost Yamabe Solitons and Gradient Almost Yamabe Solitons with Conformal Vector Fields

Ali H Alkhaldi,Pişcoran Laurian-Ioan,Akram Ali,Abimbola Abolarinwa

doi:10.3390/sym13122362

Ali H Alkhaldi, Pişcoran Laurian-Ioan + Show 2 more

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https://doi.org/10.3390/sym13122362

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Abstract

In this paper, some sufficient conditions of almost Yamabe solitons are established, such that the solitons are Yamabe metrics, by which we mean metrics of constant scalar curvature. This is achieved by imposing fewer topological constraints. The properties of the conformal vector fields are exploited for the purpose of establishing various necessary criteria on the soliton vector fields of gradient almost Yamabe solitons so as to obtain Yamabe metrics.

Highlights

Hamilton [1] on a Riemannian manifold Mn of dimension n, in which the given family of time-dependent metrics g(t) is warped by development according to the governing equation
It should be recalled that a smooth vector field ζ on (M, g) is said to be a conformal if the local flow of ζ is composed of local conformal transformations of (M, g), which commensurate with the notion that the vector field ζ is adequate; that is, L g = βg, 2 ζ where β lies in C ∞ (M) and is called the potential function of U
A vector field created by the gradient of the height function from the immersion has already proven to be a rich source for producing examples of soliton fields

Summary

Characterization of Almost

The notion of Yamabe flow was established by R. It should be noted that the soliton will be called trivial if f is a constant function on M, or the vector field ζ or ∇ f is Killing. Almost Yamabe solitons serve as particular solutions to (1) in [1] and they are fixed points of the Yamabe flow modulo diffeomorphisms and scalings of Riemannian metrics. Finding conditions on Yamabe solitons’ vector fields so that their metrics possess constant scalar curvature is one of the most fascinating topics in Yamabe solitons’ geometry. For such a study, there are two options: imposing stronger topological limits with smaller geometric and analytic conditions, or imposing reduced topological restrictions with more analytic and geometric restrictions. On the vector fields of the Yamabe solitons, we prove numerous new sufficient conditions for their metrics to be Yamabe

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Conclusions