On Killing tensors on Riemannian symmetric spaces

TL;DR

This paper classifies quadratic Killing tensor fields on Riemannian symmetric spaces, showing they are spanned by specific indecomposable and decomposable tensors, especially on rank one spaces like quaternionic and Cayley projective spaces.

Contribution

It reduces the study of Killing tensors on symmetric spaces to compact irreducible cases and provides an explicit description of top slot Killing tensor fields.

Findings

Quadratic Killing tensor fields are spanned by top-slot ones.

On quaternionic and Cayley projective spaces, they are spanned by indecomposable and decomposable tensors.

Complete classification of quadratic Killing tensors on rank one symmetric spaces.

Abstract

A Killing tensor field on a Riemannian space corresponds to an integral of the geodesic flow polynomial in momenta. A Killing tensor field is called decomposable if it is a polynomial in Killing vector fields. In this paper, we first prove that the study of Killing tensor fields on symmetric spaces can be reduced to the case of compact irreducible ones. Then we introduce the class of top slot Killing tensor fields. We obtain an explicit and elegant description of such tensor fields and prove that the quadratic Killing tensor fields are spanned by the top-slot ones. We also show that quadratic Killing tensor fields on the quaternionic projective space and on the Cayley projective space are spanned by the indecomposable ones constructed in our earlier paper and the decomposable ones. This completes the classification of quadratic Killing tensor fields on Riemannian symmetric spaces of rank one.