
Killing Tensor Fields of Third Rank on a Two-Dimensional Riemannian Torus
Author(s) -
V. A. Sharafutdinov
Publication year - 2021
Publication title -
journal of mathematics research
Language(s) - English
Resource type - Journals
eISSN - 1916-9809
pISSN - 1916-9795
DOI - 10.5539/jmr.v14n1p1
Subject(s) - mathematics , killing vector field , torus , lambda , rank (graph theory) , mathematical physics , covariant derivative , tensor (intrinsic definition) , field (mathematics) , vector field , mathematical analysis , combinatorics , pure mathematics , physics , geometry , quantum mechanics
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= λ(z)|dz|^2 in the coordinates. The torus admits a third rank Killing tensor field if and only if the function λ satisfies the equation R(∂/∂zλ(c∆^-1λ_zz+a))= 0 with some complex constants a and c≠0. The latter equation is equivalent to some system of quadratic equations relating Fourier coefficients of the function λ. If the functions λ and λ + λ_0 satisfy the equation for a real constant λ0, 0, then there exists a non-zero Killing vector field on the torus.