Killing tensor fields on the 2-torus

TL;DR

This paper investigates the existence of Killing tensor fields on the 2-torus, providing necessary conditions related to closed geodesics and Gaussian curvature for such tensor fields of arbitrary rank and specifically rank 3.

Contribution

It establishes two necessary conditions for Riemannian metrics on the 2-torus to admit Killing tensor fields, advancing understanding of their geometric properties.

Findings

Necessary conditions involving closed geodesics.

Conditions related to Gaussian curvature isolines.

Results applicable to Killing tensor fields of any rank and specifically rank 3.

Abstract

A symmetric tensor field on a Riemannian manifold is called Killing field if the symmetric part of its covariant derivative is equal to zero. There is a one to one correspondence between Killing tensor fields and first integrals of the geodesic flow which depend polynomially on the velocity. Therefore Killing tensor fields closely relate to the problem of integrability of geodesic flows. In particular, the following question is still open: does there exist a Riemannian metric on the 2-torus which admits an irreducible Killing tensor field of rank $\geq 3$? We obtain two necessary conditions on a Riemannian metric on the 2-torus for the existence of Killing tensor fields. The first condition is valid for Killing tensor fields of arbitrary rank and relates to closed geodesics. The second condition is obtained for rank 3 Killing tensor fields and relates to isolines of the Gaussian curvature.