Killing tensors on tori

TL;DR

This paper proves that on certain conformally flat tori, all polynomial first integrals of the geodesic flow are linear in momenta, showing a specific structure of Killing tensors in this setting.

Contribution

It characterizes Killing tensors on conformally flat tori with one-variable-dependent conformal factors as polynomials in the metric and Killing vectors, revealing their algebraic structure.

Findings

Killing tensors are polynomials in the metric and Killing vectors.

First integrals polynomial in momenta are linear in momenta.

Results apply to conformally flat tori with one-variable-dependent conformal factors.

Abstract

We show that Killing tensors on conformally flat $n$-dimensional tori whose conformal factor only depends on one variable, are polynomials in the metric and in the Killing vector fields. In other words, every first integral of the geodesic flow polynomial in the momenta on the sphere bundle of such a torus is linear in the momenta.