浸入曲线的邻域与流形

IF 1 Q1 MATHEMATICS

INTERNATIONAL JOURNAL OF MATHEMATICS AND MATHEMATICAL SCIENCES Pub Date : 2021-12-31 DOI:10.1155/2021/6974292

A. Mennucci

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引用次数: 0

摘要

我们给出了流形间浸入式的一些精细性质，特别注意浸入式曲线c: s1 ?我们提出了新的结果，以及已知的结果，但有定量陈述(可能在数值应用中有用)，涉及管状坐标，浸入和自由浸入曲线的邻域，以及附近此类曲线的局部唯一表示，可能“直至重新参数化”。我们给出了例子和反例来支持这些结果的重要性。最后，我们提供了Cervera, Mascaró和Michor在1991年的一篇论文中首次提出的一个结果的完整和详细的证明:由重参数化作用的自由浸入曲线的商是一个光滑(无限维)流形。

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Neighborhoods and Manifolds of Immersed Curves

We present some fine properties of immersions ℐ : M ⟶ N between manifolds, with particular attention to the case of immersed curves c : S 1 ⟶ ℝ n . We present new results, as well as known results but with quantitative statements (that may be useful in numerical applications) regarding tubular coordinates, neighborhoods of immersed and freely immersed curve, and local unique representations of nearby such curves, possibly “up to reparameterization.” We present examples and counterexamples to support the significance of these results. Eventually, we provide a complete and detailed proof of a result first stated in a 1991-paper by Cervera, Mascaró, and Michor: the quotient of the freely immersed curves by the action of reparameterization is a smooth (infinite dimensional) manifold.

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来源期刊

INTERNATIONAL JOURNAL OF MATHEMATICS AND MATHEMATICAL SCIENCES Mathematics-Mathematics (miscellaneous)

CiteScore

2.30

自引率

8.30%

发文量

审稿时长

17 weeks

期刊介绍： The International Journal of Mathematics and Mathematical Sciences is a refereed math journal devoted to publication of original research articles, research notes, and review articles, with emphasis on contributions to unsolved problems and open questions in mathematics and mathematical sciences. All areas listed on the cover of Mathematical Reviews, such as pure and applied mathematics, mathematical physics, theoretical mechanics, probability and mathematical statistics, and theoretical biology, are included within the scope of the International Journal of Mathematics and Mathematical Sciences.