Ricci Solitons, Almost Theta-Yamabe Solitons, and Finite-Order Tensor Symmetries of a Vector Field on Riemannian Manifolds with Rank-One Anisotropic Curvature

Abstract

We study the iterated action of the Lie derivative on the curvature tensor of a Riemannian manifold with rank-one anisotropic curvature, whose Riemann tensor is expressed via the Kulkarni-Nomizu product as $R = λ(ξ^\flat\otimesξ^\flat)\owedge g$. First, we examine the conditions under which such a manifold admits a Ricci soliton structure and demonstrate that this property implies the almost $θ$-Yamabe soliton structure. Furthermore, we show that if the associated potential vector field $X$ is a symmetry of the Ricci tensor of a fixed order $k$ (i.e., $\mathcal{L}_X^k \operatorname{Ric} = 0$), the geometric problem reduces to solving a partial differential equation of order $k+1$ along the flow. Finally, under the assumption that $X$ is a conformal vector field ($\mathcal{L}_X g = 2φg$) whose infinitesimal flow preserves the line distribution $\mathcal{D}=\operatorname{Span}\{ξ\}$ (with $[X,ξ]=aξ$ for $a\in\mathbb{R}$), we prove that several key geometric problems (such as establishing the relation $\mathcal{L}_X^k R = R$, determining the minimal order $k$ for $X$ to be a Lie curvature symmetry, or satisfying $\mathcal{L}_X^{k+1}R = f \mathcal{L}_X^k R$ for a continuous function $f$) are equivalent to a scalar differential problem governed by the operator $D_X = X + 6φ+ 2a$.