Zeitlin's model is a spatial discretization for the 2-D Euler equations on the flat 2-torus or the 2-sphere. Contrary to other discretizations, it preserves the underlying geometric structure, namely that the Euler equations describe Riemannian geodesics on a Lie group. Here we show how to extend Zeitlin's approach to the axisymmetric Euler equations on the 3-sphere. It is the first discretization of the 3-D Euler equations that fully preserves the geometric structure, albeit restricted to axisymmetric solutions. Thus, this finite-dimensional model admits Riemannian curvature and Jacobi equations, which are discussed.

翻译：Zeitlin模型是平直二维环面或二维球面上二维欧拉方程的一种空间离散化方法。与其他离散化方法不同，该模型保持了底层几何结构，即欧拉方程描述了李群上的黎曼测地线。本文展示了如何将Zeitlin方法推广至三维球面上的轴对称欧拉方程。这是首个完全保持几何结构的三维欧拉方程离散化模型（尽管仅限于轴对称解）。因此，该有限维模型允许存在黎曼曲率和雅可比方程，本文对此进行了讨论。