This article is cited in 6 papers

Killing tensor fields on the $2$-torus

V. A. Sharafutdinov<sup>ab

a</sup> Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
^b Novosibirsk State University, Novosibirsk, Russia

Abstract: A symmetric tensor field on a Riemannian manifold is called a Killing field if the symmetric part of its covariant derivative equals 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 relate closely 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 $\ge3$? 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 pertains to isolines of the Gaussian curvature.

Keywords: Killing fields, integrability of geodesic flows, tensor analysis, the method of spherical harmonics.

UDC: 514.764.2+517.954

Received: 17.11.2014

DOI: 10.17377/smzh.2016.57.115

 English version: Siberian Mathematical Journal, 2016, 57:1, 155–173

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