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♭⊗ξ♭)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., ℒXk 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 (ℒX g = 2φg) whose infinitesimal flow preserves the line distribution 𝒟=Span{ξ} (with [X,ξ]=aξ for a∈ℝ), we prove that several key geometric problems (such as establishing the relation ℒXk R = R, determining the minimal order k for X to be a Lie curvature symmetry, or satisfying ℒXk+1R = f ℒXk R for a continuous function f) are equivalent to a scalar differential problem governed by the operator DX = X + 6φ+ 2a.
Related
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BibTeX
@misc{bousso2026ricci,
title = {{Ricci Solitons, Almost Theta-Yamabe Solitons, and Finite-Order Tensor Symmetries of a Vector Field on Riemannian Manifolds with Rank-One Anisotropic Curvature}},
author = {Abdou Bousso and Ameth Ndiaye},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14866},
eprint = {2609.14866},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14866},
}
RIS
TY - UNPB
TI - Ricci Solitons, Almost Theta-Yamabe Solitons, and Finite-Order Tensor Symmetries of a Vector Field on Riemannian Manifolds with Rank-One Anisotropic Curvature
AU - Abdou Bousso
AU - Ameth Ndiaye
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14866
UR - https://arxiv.org/abs/2609.14866
AB - 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^♭⊗ξ^♭)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., ℒ_X^k 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 (ℒ_X g = 2φg) whose infinitesimal flow preserves the line distribution 𝒟=Span{ξ} (with [X,ξ]=aξ for a∈ℝ), we prove that several key geometric problems (such as establishing the relation ℒ_X^k R = R, determining the minimal order k for X to be a Lie curvature symmetry, or satisfying ℒ_X^k+1R = f 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.
ER -