Killing Tensor Fields of Third Rank on a Two-Dimensional Riemannian Torus

V A Sharafutdinov

doi:10.5539/jmr.v14n1p1

Abstract

A rank m symmetric tensor field on a Riemannian manifold is called a Killing field if the symmetric part of its covariant derivative is equal to zero.  Such a field determines the first integral of the geodesic flow which is a degree m homogeneous polynomial in velocities. There exist global isothermal coordinates on a two-dimensional Riemannian torus such that the metric is of the form ds^2= &lambda;(z)|dz|^2 in the coordinates. The torus admits a third rank Killing tensor field if and only if the function &lambda; satisfies the equation R(&part;/&part;z（&lambda;(c∆^-1&lambda;_zz+a))= 0 with some complex constants a and c&ne;0. The latter equation is equivalent to some system of quadratic equations relating Fourier coefficients of the function &lambda;. If the functions &lambda; and &lambda; + &lambda;_0 satisfy the equation for a real constant &lambda;0, 0, then there exists a non-zero Killing vector field on the torus.

Highlights

We first recall the definition of a Killing tensor field on an arbitrary Riemannian manifold.Given a Riemannian manifold (M, g), let S mτM be the bundle of rank m symmetric tensors
There exist global isothermal coordinates on a two-dimensional Riemannian torus such that the metric is of the form ds2 = λ(z)|dz|2 in the coordinates
The space C∞(S m) of smooth sections of the bundle consists of rank m symmetric tensor fields

Summary

Introduction

We first recall the definition of a Killing tensor field on an arbitrary Riemannian manifold. A Riemannian torus (C/Γ, λ|dz|2) admits a non-trivial rank 3 Killing tensor field if and only if Fourier coefficients of the function λ satisfy the equations k∈Γ c1(−n1k12 +2n2k1k2 +n1k22). The geometric sense of such solutions is obvious: If the equation (8) possesses a Γ-periodic one-dimensional solution, the metric can be transformed to the form (2)–(3) by some rotation of the lattice Γ and the torus admits a non-trivial Killing vector field f. In such a case f 3 = f ⊗ f ⊗ f is the reducible Killing tensor field of third rank. Last two sections are devoted to the proof of Theorem 5 which turns out to be quite difficult

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 5

Proof of Lemma 7

Conclusion