Killing and Conformal Killing tensors

TL;DR

This paper develops a formalism for conformal Killing tensors on Riemannian manifolds, rederives known results, and presents new classifications, non-existence results, and examples of such tensors.

Contribution

It introduces a new formalism for conformal Killing tensors, simplifies existing proofs, and provides novel classifications and examples in the field.

Findings

Classification of conformal Killing 2-tensors on Riemannian products

Weitzenböck formulas leading to non-existence results

Construction of manifolds with conformal Killing tensors

Abstract

We introduce an appropriate formalism in order to study conformal Killing (symmetric) tensors on Riemannian manifolds. We reprove in a simple way some known results in the field and obtain several new results, like the classification of conformal Killing $2$-tensors on Riemannian products of compact manifolds, Weitzenb\"ock formulas leading to non-existence results, and construct various examples of manifolds with conformal Killing tensors.