On the geometry of curves and conformal geodesics in the Mobius space
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2010
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| _version_ | 1866913543627997184 |
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| author | Magliaro, Marco Mari, Luciano Rigoli, Marco |
| author_facet | Magliaro, Marco Mari, Luciano Rigoli, Marco |
| contents | This paper deals with the study of some properties of immersed curves in the conformal sphere $\mathds{Q}_n$, viewed as a homogeneous space under the action of the Möbius group. After an overview on general well-known facts, we briefly focus on the links between Euclidean and conformal curvatures, in the spirit of F. Klein's Erlangen program. The core of the paper is the study of conformal geodesics, defined as the critical points of the conformal arclength functional. After writing down their Euler-Lagrange equations for any $n$, we prove an interesting codimension reduction, namely that every conformal geodesic in $\mathds{Q}_n$ lies, in fact, in a totally umbilical 4-sphere $\mathds{Q}_4$. We then extend and complete the work in (Musso, "The conformal arclength functional", Math Nachr.) by solving the Euler-Lagrange equations for the curvatures and by providing an explicit expression even for those conformal geodesics not included in any conformal 3-sphere. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1003_5776 |
| institution | arXiv |
| publishDate | 2010 |
| record_format | arxiv |
| spellingShingle | On the geometry of curves and conformal geodesics in the Mobius space Magliaro, Marco Mari, Luciano Rigoli, Marco Differential Geometry primary 53A30, secondary 53C30 This paper deals with the study of some properties of immersed curves in the conformal sphere $\mathds{Q}_n$, viewed as a homogeneous space under the action of the Möbius group. After an overview on general well-known facts, we briefly focus on the links between Euclidean and conformal curvatures, in the spirit of F. Klein's Erlangen program. The core of the paper is the study of conformal geodesics, defined as the critical points of the conformal arclength functional. After writing down their Euler-Lagrange equations for any $n$, we prove an interesting codimension reduction, namely that every conformal geodesic in $\mathds{Q}_n$ lies, in fact, in a totally umbilical 4-sphere $\mathds{Q}_4$. We then extend and complete the work in (Musso, "The conformal arclength functional", Math Nachr.) by solving the Euler-Lagrange equations for the curvatures and by providing an explicit expression even for those conformal geodesics not included in any conformal 3-sphere. |
| title | On the geometry of curves and conformal geodesics in the Mobius space |
| topic | Differential Geometry primary 53A30, secondary 53C30 |
| url | https://arxiv.org/abs/1003.5776 |