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:S1⟶ℝn. We present new results, as well as known results but with quantitative statements (that may be useful in numerical applications) regarding tubular coordinates, nei...
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| Format: | Article |
| Language: | English |
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Hindawi Limited
2021-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2021/6974292 |
| _version_ | 1798034281351610368 |
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| author | Andrea C. G. Mennucci |
| author_facet | Andrea C. G. Mennucci |
| author_sort | Andrea C. G. Mennucci |
| collection | DOAJ |
| description | We present some fine properties of immersions ℐ:M⟶N between manifolds, with particular attention to the case of immersed curves c:S1⟶ℝn. 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. |
| first_indexed | 2024-04-11T20:42:08Z |
| format | Article |
| id | doaj.art-e9ccff826d7a407eaeee42a511e78749 |
| institution | Directory Open Access Journal |
| issn | 1687-0425 |
| language | English |
| last_indexed | 2024-04-11T20:42:08Z |
| publishDate | 2021-01-01 |
| publisher | Hindawi Limited |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj.art-e9ccff826d7a407eaeee42a511e787492022-12-22T04:04:11ZengHindawi LimitedInternational Journal of Mathematics and Mathematical Sciences1687-04252021-01-01202110.1155/2021/6974292Neighborhoods and Manifolds of Immersed CurvesAndrea C. G. Mennucci0Scuola Normale SuperioreWe present some fine properties of immersions ℐ:M⟶N between manifolds, with particular attention to the case of immersed curves c:S1⟶ℝn. 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.http://dx.doi.org/10.1155/2021/6974292 |
| spellingShingle | Andrea C. G. Mennucci Neighborhoods and Manifolds of Immersed Curves International Journal of Mathematics and Mathematical Sciences |
| title | Neighborhoods and Manifolds of Immersed Curves |
| title_full | Neighborhoods and Manifolds of Immersed Curves |
| title_fullStr | Neighborhoods and Manifolds of Immersed Curves |
| title_full_unstemmed | Neighborhoods and Manifolds of Immersed Curves |
| title_short | Neighborhoods and Manifolds of Immersed Curves |
| title_sort | neighborhoods and manifolds of immersed curves |
| url | http://dx.doi.org/10.1155/2021/6974292 |
| work_keys_str_mv | AT andreacgmennucci neighborhoodsandmanifoldsofimmersedcurves |