论文标题
对称矩阵的单模型家族
Unimodular families of symmetric matrices
论文作者
论文摘要
我们介绍了对称矩阵值的映射件之间的体积保留等效性，这是布鲁斯的$ \ Mathcal {g} $ - 等价的单模型版本。相对于$ \ Mathcal {g} $ - 等效性从分类中推断出单模型分类的关键概念是对称的准习惯性，这是对$ 2 \ times 2 $对称矩阵估计的$ 2 $ symmetric matrix valued map-germ的条件的概括。如果一个$ \ MATHCAL {G} $ - 等价类包含一个对称的准同质代表，则类相对于体积扩张等效性而言，该类与该类别相一致（如果地面场是真实的，则相对于音量扩张等效性（以防定向逆转差异性）。通过使用该方法，我们表明，与$ \ Mathcal {G} $相对于Bruce列表中的所有简单类别的等效性与相对于卷保持等价的相关。然后，我们将MAP-GERS从飞机分类为$ 2 \ times 2 $和$ 3 \ times 3 $最多的$ 1 $的真实对称矩阵，$ \ Mathcal {g} _e $ sodimension小于$ 9 $，我们显示的一些正常形式分为两种不同的单型单模型的单数。我们提供了几个示例，以说明非简单性并不意味着非对称的准准习惯性，并且MAP-GERM是对称的准综合性的条件要比MAP-GERM的每个组件都更强，而MAP-GERM的每个组成部分都是准同质的。我们还提供了一个相对于$ \ Mathcal {G} $及其相应的形式的非模块的正常形式的非对称准同质正常形式的示例。
We introduce the volume-preserving equivalence among symmetric matrix-valued map-germs which is the unimodular version of Bruce's $\mathcal{G}$-equivalence. The key concept to deduce unimodular classification out of classification relative to $\mathcal{G}$-equivalence is symmetrical quasi-homogeneity, which is a generalization of the condition for a $2 \times 2$ symmetric matrix-valued map-germ in Corollary~2.1 (ii) by Bruce, Goryunov and Zakalyukin. If a $\mathcal{G}$-equivalence class contains a symmetrically quasi-homogeneous representative, the class coincides with that relative to the volume-preserving equivalence (up to orientation reversing diffeomorphism in case if the ground field is real). By using that we show that all the simple classes relative to $\mathcal{G}$-equivalence in Bruce's list coincides with those relative to the volume preserving equivalence. Then, we classify map-germs from the plane to the set of $2 \times 2$ and $3 \times 3$ real symmetric matrices of corank at most $1$ and of $\mathcal{G}_e$-codimension less than $9$ and we show some of the normal forms split into two different unimodular singularities. We provide several examples to illustrate that non simplicity does not imply non symmetrical quasi-homogeneity and the condition that a map-germ is symmetrically quasi-homogeneous is stronger than one that each component of the map-germ is quasi-homogeneous. We also present an example of non symmetrically quasi-homogeneous normal form relative to $\mathcal{G}$ and its corresponding formal unimodular normal form.
