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| Príomhchruthaitheoirí: | , , |
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| Formáid: | Preprint |
| Foilsithe / Cruthaithe: |
2019
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| Ábhair: | |
| Rochtain ar líne: | https://arxiv.org/abs/1902.02235 |
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| _version_ | 1866914047483445248 |
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| author | Brasselet, Jean-paul Ruas, Maria Aparecida Soares Nguyen, Thuy |
| author_facet | Brasselet, Jean-paul Ruas, Maria Aparecida Soares Nguyen, Thuy |
| contents | We provide bi-Lipschitz invariants for finitely determined map germs $f: (\mathbb{K}^n,0) \to (\mathbb{K}^p, 0)$, where $\mathbb{K} = \mathbb{R}$ or $ \mathbb{C}$. The aim of the paper is to provide partial answers to the following questions:
Does the bi-Lipschitz type of a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$ determine the bi-Lipschitz type of the link of $f$ and of the double point set of $f$? Reciprocally, given a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$, do the bi-Lipschitz types of the link of $f$ and of the double point set of $f$ determine the bi-Lipschitz type of the germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$?
We provide a positive answer to the first question in the case of a finitely determined map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$ where $2n-1 \leq p$ (Theorem 3.3). With regard to the second question, for a finitely determined map germ $f : (\mathbb{R}^2,0) \to (\mathbb{R}^3,0),$ we show that a complete set of invariants for the bi-Lipschitz classification with respect to the inner metric of $X_f=f(U)$, where $U$ is a small neighbourhood of the origin in $\mathbb R^2$, is is given by the link of $f$, the image of the double point set of $f$ and the polar curve of a generic projection into the plane (Proposition 4.13). In particular, in the homogeneous parametrization case $f: (\mathbb{R}^2, 0) \to (\mathbb{R}^3, 0)$ of corank 1, we do not need the hypothesis on the equivalence of the image of the double point set (Theorem 5.2). Finally, we apply our results to relate the $C^{0}- \mathcal A$ classes of finitely determined map germs $f$ of corank 1 with homogeneous parametrization and the inner bi-Lipschitz type of $X_f$ (Proposition 5.4). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1902_02235 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Local bi-Lipschitz classification of semialgebraic surfaces Brasselet, Jean-paul Ruas, Maria Aparecida Soares Nguyen, Thuy Geometric Topology We provide bi-Lipschitz invariants for finitely determined map germs $f: (\mathbb{K}^n,0) \to (\mathbb{K}^p, 0)$, where $\mathbb{K} = \mathbb{R}$ or $ \mathbb{C}$. The aim of the paper is to provide partial answers to the following questions: Does the bi-Lipschitz type of a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$ determine the bi-Lipschitz type of the link of $f$ and of the double point set of $f$? Reciprocally, given a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$, do the bi-Lipschitz types of the link of $f$ and of the double point set of $f$ determine the bi-Lipschitz type of the germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$? We provide a positive answer to the first question in the case of a finitely determined map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$ where $2n-1 \leq p$ (Theorem 3.3). With regard to the second question, for a finitely determined map germ $f : (\mathbb{R}^2,0) \to (\mathbb{R}^3,0),$ we show that a complete set of invariants for the bi-Lipschitz classification with respect to the inner metric of $X_f=f(U)$, where $U$ is a small neighbourhood of the origin in $\mathbb R^2$, is is given by the link of $f$, the image of the double point set of $f$ and the polar curve of a generic projection into the plane (Proposition 4.13). In particular, in the homogeneous parametrization case $f: (\mathbb{R}^2, 0) \to (\mathbb{R}^3, 0)$ of corank 1, we do not need the hypothesis on the equivalence of the image of the double point set (Theorem 5.2). Finally, we apply our results to relate the $C^{0}- \mathcal A$ classes of finitely determined map germs $f$ of corank 1 with homogeneous parametrization and the inner bi-Lipschitz type of $X_f$ (Proposition 5.4). |
| title | Local bi-Lipschitz classification of semialgebraic surfaces |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/1902.02235 |