In many problems in analysis and geometry there is a need to investigate vector fields with singular points that are not isolated but rather form a submanifold of the phase space, which most often has codimension 2. Of primary interest are the local orbital normal forms of such fields. ‘Orbital’ means that we may multiply vector fields by scalar functions with constant sign. In what follows, all vector fields and functions are assumed without mention to be smooth (of class C∞) unless otherwise stated. Roussarie [1] investigated vector fields of a certain special type which satisfy the following conditions at all singular points: 1) the components of the field lie in the ideal (of the space of smooth functions) generated by two of the components; 2) the divergence of the vector field (the trace of its linear part) is zero. We call such fields R-fields after Roussarie. In local coordinates the germ of an R-field has the following form at its singular point:

中文翻译：

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具有退化二次部分的双曲 Roussarie 场

在分析和几何中的许多问题中，需要研究具有奇异点的矢量场，这些奇异点不是孤立的，而是形成相空间的子流形，通常具有余维数 2。主要感兴趣的是此类的局部轨道法线形式字段。“轨道”意味着我们可以将向量场乘以具有常数符号的标量函数。在下文中，除非另有说明，否则假定所有向量场和函数都是平滑的（属于 C∞ 类）。Roussarie [1] 研究了在所有奇异点处满足以下条件的某种特殊类型的矢量场： 1）场的分量位于由两个分量生成的理想（光滑函数空间）中；2）矢量场的散度（其线性部分的迹线）为零。我们在 Roussarie 之后将此类场称为 R 场。在局部坐标中，R 场的胚芽在其奇点处具有以下形式：