2025/12/10 by С. А. Степанов, Stepanov, Sergey, Irina Tsyganok +1
Mathematics · #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2512.09605
openalex publication_date 2025/12/10 · openalex created_date 2025/12/12 · openalex updated_date 2026/07/30
First-order differential operators arising from the representation-theoretic decomposition of the covariant derivative play a central role in Riemannian geometry. In this paper, we study Stein-Weiss O(n)-gradients acting on covariant symmetric trace-free tensors of arbitrary rank p ≥ 2. By analyzing the decomposition of T^*M ⊗ S0p(M) into its O(n)-irreducible components, we explicitly describe the corresponding generalized gradients and compute Weitzenbock formulas for their adjoint compositions. These results extend Bouguignon four-dimensional formulas for p = 2 and generalize previous work of other authors to higher-rank symmetric tensors. The formulas obtained provide a unified framework for understanding second-order Stein-Weiss operators and yield tools applicable to deformation complexes, curvature estimates, and stability problems in geometric analysis. The article continues the authors' earlier investigations of Stein-Weiss operators on natural tensor bundles.