Extensions and Applications of Stein-Weiss Operators to the Study of Traceless Symmetric Tensors

TL;DR

This paper extends Stein-Weiss operators to higher-rank traceless symmetric tensors, providing explicit decompositions, generalized gradients, and Weitzenbock formulas with applications in geometric analysis.

Contribution

It introduces a comprehensive framework for Stein-Weiss operators on symmetric trace-free tensors of any rank, generalizing previous results and enabling new applications.

Findings

Explicit decomposition of tensor bundles into irreducible components

Generalized gradients and Weitzenbock formulas derived

Framework applicable to deformation, curvature, and stability analyses

Abstract

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 $p \ge 2$. By analyzing the decomposition of $T^*M \otimes S_0^p(M)$ 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.