Ricci-type symmetries and Ricci solitons on two-dimensional Riemannian manifolds with diagonal metrics in local orthogonal coordinates

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Abstract

This paper develops a unified framework for infinitesimal geometric symmetries and Ricci- type structures on two-dimensional Riemannian manifolds represented locally in orthogonal coordi- nates by a diagonal metric. Using a local orthonormal frame adapted to the metric coefficients, we derive explicit expressions for the Riemannian curvature tensor and the Ricci tensor in terms of the associated structure functions. Since, in dimension two, the Ricci tensor satisfies the pointwise iden- tity Ric =Kg, the symmetry conditions considered can be reduced to systems of differential equa- tions involving the Gaussian curvature, the metric coefficients, and the components of a vector field V =V¹ E₁ +V² E₂. We obtain necessary and sufficient conditions for conformal vector fields, ρ–Ricci vector fields, Ricci collineations, conformal Ricci collineations, and Ricci solitons and analyze the resulting systems under natural separability and cross-dependence assumptions. This leads to explicit solution families, rigidity results, and curvature restrictions for the corresponding symmetry classes. In the flat cases arising from the separability assumptions, the underlying metric is locally Euclidean, whereas, on connected open regions where the Gaussian curvature is nowhere zero, the corresponding symmetry equations retain an essential curvature dependence. In particular, we characterize the sep- arable case, determine the conditions under which the orthonormal frame fields generate Ricci-type symmetries, obtain explicit formulas for conformal Ricci collineations, and derive necessary and suffi- cient conditions for Ricci solitons. For ρ–Ricci vector fields, the classification of the diagonal metrics for which the coordinate vector fields are ρ–Ricci is carried out under the additional assumption that ρ is a prescribed nonzero constant, yielding explicit differential constraints on the metric coefficients. Several examples are provided to illustrate the existence of non-Killing conformal fields and Ricci soliton structures. The results are local in nature and apply within the orthogonal coordinate setting and the coordinate-dependence assumptions considered in this paper.

Keywords

two-dimensional Riemannian manifold / diagonal metric / conformal vector field / ρ–Ricci vector fields / Riemannian curvature tensor / Ricci tensor

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