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

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 $R=\lambda ({\xi }^{♭}\otimes {\xi }^{♭})\text{owedge}g$. First, we examine the conditions under which such a manifold admits a Ricci soliton structure and demonstrate that this property implies the almost $\theta$-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., ${\mathcal{L}}_X^k\mathrm{Ric}=0$), 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 (${\mathcal{L}}_{X}g=2\phi g$) whose infinitesimal flow preserves the line distribution $\mathcal{D}=\mathrm{Span}\{\xi \}$ (with $[X,\xi ]=a\xi$ 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 $D_X=X+6\phi +2a$.