Mathematics Interdisciplinary Research

Mathematics Interdisciplinary Research

$(\alpha,\beta)$-Ricci-Yamabe Solitons with Potential Kuiper-Nash Vector Field

Document Type : Original Scientific Paper

Authors
Shahroud azami 1
Mehdi Jafari 2
Mosayeb Zohrehvand 3
1 ‎Department of Pure Mathematics‎, ‎‎Faculty of Science, Imam Khomeini International University, Qazvin‎, ‎I‎. ‎R‎. ‎Iran
2 ‎Department of Mathematics, ‎Payame Noor University, ‎Tehran‎, ‎I‎. ‎R‎. ‎Iran
3 ‎Department of Mathematics‎, ‎ ‎Faculty of Mathematical Sciences and Statistics, ‎Malayer University, ‎Malayer‎, ‎I‎. ‎R‎. ‎Iran
10.22052/mir.2026.258097.1561
Abstract
‎
‎This study focuses on identifying conditions that allow an $n$-dimensional compact Riemannian manifold $(M^n‎, ‎g)$ to support a vector field $\xi$ along with smooth functions $\alpha$ and $\lambda$‎, forming an $(\alpha,\beta)$-Ricci--Yamabe soliton structure. Our construction relies on embedding $(M^n‎, ‎g)$ isometrically into the Euclidean space $(\mathbb{R}^m‎, ‎g)$ following the method of Kuiper and Nash‎, ‎where $\xi$ is defined as the tangential component of a fixed unit vector in $\mathbb{R}^m$‎, ‎hereafter called the Kuiper-Nash vector‎. Let $r$ denote the scalar curvature of $(M^n‎, ‎g)$, ‎we establish that whenever the integral of $\xi(r)$ satisfies a specific lower bound dependent on a constant $\lambda$‎, ‎the tuple $(M^n‎, ‎g‎, ‎\beta r‎, ‎\xi‎, ‎\lambda)$ forms a valid $(\alpha,\beta)$-Ricci-Yamabe soliton‎, ‎which we term a Kuiper-Nash $(\alpha,\beta)$-Ricci-Yamabe soliton‎. ‎We examine conditions under which the soliton becomes trivial‎, ‎presenting a characterization that directly relates this behavior to the scalar curvature r‎.
‎Finally‎, ‎an explicit characterization is derived for a compact‎, ‎n-dimensional trivial Kuiper-Nash $(\alpha,\beta)$-Ricci-Yamabe soliton‎, ‎expressed in terms of an upper bound on the integral of $(\mathrm{div}\,\xi)^2$ that incorporates r.
Keywords
Kuiper-Nash embedding&lrm
Generalized $\mathcal{Z}$-soliton&lrm
, &lrm
Shape operator
Subjects
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Mathematics Interdisciplinary Research
Volume 11, Issue 4
In Progress
Autumn 2026
Pages 351-370