论文标题
在二维Riemannian torus中杀死第三排名的张量
Killing tensor fields of third rank on a two-dimensional Riemannian torus
论文作者
论文摘要
如果其协变量的对称部分等于零，则在Riemannian歧管上的等级$ M $对称张量场被称为杀戮场。这样的磁场决定了地球流的第一个积分，该流量是$ m $均质的速度多项式。在二维Riemannian圆环上存在全局等温坐标，因此坐标中的度量标准为$ ds^2 =λ（z）| dz |^2 $。当且仅当功能$λ$满足方程$ \ re \ big（\ frac {\ partial} {\ partial z} \ big（λ（cΔ^^{ - 1}λ_ /λ_）λ_{ZZ}+a）\ big）= 0 $ a $时，后一个方程相当于某些二次方程系统，与函数$λ$的傅立叶系数相关。如果功能$λ$和$λ+λ_0$满足真实常数$λ_0\ neq0 $的方程，则在圆环上存在非零杀伤向量场。
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 $\Re\big(\frac{\partial}{\partial z}\big(λ(cΔ^{-1}λ_{zz}+a)\big)\big)=0$ with some complex constants $a$ and $c\neq0$. 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 $λ_0\neq0$, then there exists a non-zero Killing vector field on the torus.
