Reparametrizations of vector fields and their shift maps
LetM be a smooth manifold, F be a smooth vector field on M, and (Ft) be the local flow of F. Denote by Sh(F) the subset of C^∞(M,M) consisting of maps h : M → M of the following form:
 h(x) = Fα(x)(x), where _ runs over all smooth functions M → R which can be substituted into F instead of t....
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| Date: | 2009 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/6328 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Reparametrizations of vector fields and their shift maps / S. Maksymenko // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 489-498. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862734175259328512 |
|---|---|
| author | Maksymenko, S. |
| author_facet | Maksymenko, S. |
| citation_txt | Reparametrizations of vector fields and their shift maps / S. Maksymenko // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 489-498. — Бібліогр.: 8 назв. — англ. |
| collection | DSpace DC |
| description | LetM be a smooth manifold, F be a smooth vector field on M, and (Ft) be the local flow of F. Denote by Sh(F) the subset of C^∞(M,M) consisting of maps h : M → M of the following form:
h(x) = Fα(x)(x), where _ runs over all smooth functions M → R which can be substituted into F instead of t. This space often contains the identity component of the group of diffeomorphisms preserving orbits of F. In this note it is shown that Sh(F) is not changed under reparametrizations of F, that is for any smooth strictly positive function μ : M → (0,+∞) we have that Sh(F) = Sh(μF). As an application it is proved that F can be reparametrized to induce a circle action on M if and only if there exists a smooth function μ : M → (0,+∞) such that F(x, μ(x)) ≡ x.
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| first_indexed | 2025-12-07T19:41:28Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-6328 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-2910 |
| language | English |
| last_indexed | 2025-12-07T19:41:28Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Maksymenko, S. 2010-02-23T14:50:10Z 2010-02-23T14:50:10Z 2009 Reparametrizations of vector fields and their shift maps / S. Maksymenko // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 489-498. — Бібліогр.: 8 назв. — англ. 1815-2910 https://nasplib.isofts.kiev.ua/handle/123456789/6328 LetM be a smooth manifold, F be a smooth vector field on M, and (Ft) be the local flow of F. Denote by Sh(F) the subset of C^∞(M,M) consisting of maps h : M → M of the following form:
 h(x) = Fα(x)(x), where _ runs over all smooth functions M → R which can be substituted into F instead of t. This space often contains the identity component of the group of diffeomorphisms preserving orbits of F. In this note it is shown that Sh(F) is not changed under reparametrizations of F, that is for any smooth strictly positive function μ : M → (0,+∞) we have that Sh(F) = Sh(μF). As an application it is proved that F can be reparametrized to induce a circle action on M if and only if there exists a smooth function μ : M → (0,+∞) such that F(x, μ(x)) ≡ x. en Інститут математики НАН України Геометрія, топологія та їх застосування Reparametrizations of vector fields and their shift maps Article published earlier |
| spellingShingle | Reparametrizations of vector fields and their shift maps Maksymenko, S. Геометрія, топологія та їх застосування |
| title | Reparametrizations of vector fields and their shift maps |
| title_full | Reparametrizations of vector fields and their shift maps |
| title_fullStr | Reparametrizations of vector fields and their shift maps |
| title_full_unstemmed | Reparametrizations of vector fields and their shift maps |
| title_short | Reparametrizations of vector fields and their shift maps |
| title_sort | reparametrizations of vector fields and their shift maps |
| topic | Геометрія, топологія та їх застосування |
| topic_facet | Геометрія, топологія та їх застосування |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/6328 |
| work_keys_str_mv | AT maksymenkos reparametrizationsofvectorfieldsandtheirshiftmaps |