On Singularities of the Gauss Map Components of Surfaces in R4
The Gauss map of a generic immersion of a smooth, oriented surface into R 4 is an immersion. But this map takes values on the Grassmanian of oriented 2-planes in R 4 . Since this manifold has a structure of a product of two spheres, the Gauss map has two components that take values on the sphere. We...
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| Veröffentlicht in: | The Journal of geometric analysis 2024-06, Vol.34 (6) |
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| creator | Domitrz, Wojciech Hernández-Martínez, Lucía Ivonne Sánchez-Bringas, Federico |
| description | The Gauss map of a generic immersion of a smooth, oriented surface into
R
4
is an immersion. But this map takes values on the Grassmanian of oriented 2-planes in
R
4
. Since this manifold has a structure of a product of two spheres, the Gauss map has two components that take values on the sphere. We study the singularities of the components of the Gauss map and relate them to the geometric properties of the generic immersion. Moreover, we prove that the singularities are generically stable, and we connect them to the contact type of the surface and
J
-holomorphic curves with respect to an orthogonal complex structure
J
on
R
4
. Finally, we get some formulas of Gauss–Bonnet type involving the geometry of the singularities of the components with the geometry and topology of the surface. |
| doi_str_mv | 10.1007/s12220-024-01616-7 |
| format | Article |
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R
4
is an immersion. But this map takes values on the Grassmanian of oriented 2-planes in
R
4
. Since this manifold has a structure of a product of two spheres, the Gauss map has two components that take values on the sphere. We study the singularities of the components of the Gauss map and relate them to the geometric properties of the generic immersion. Moreover, we prove that the singularities are generically stable, and we connect them to the contact type of the surface and
J
-holomorphic curves with respect to an orthogonal complex structure
J
on
R
4
. Finally, we get some formulas of Gauss–Bonnet type involving the geometry of the singularities of the components with the geometry and topology of the surface.</description><identifier>ISSN: 1050-6926</identifier><identifier>EISSN: 1559-002X</identifier><identifier>DOI: 10.1007/s12220-024-01616-7</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Abstract Harmonic Analysis ; Convex and Discrete Geometry ; Differential Geometry ; Dynamical Systems and Ergodic Theory ; Fourier Analysis ; Global Analysis and Analysis on Manifolds ; Mathematics ; Mathematics and Statistics ; Singularity (mathematics) ; Submerging ; Topology</subject><ispartof>The Journal of geometric analysis, 2024-06, Vol.34 (6)</ispartof><rights>The Author(s) 2024</rights><rights>The Author(s) 2024. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><orcidid>0000-0003-3979-7101</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s12220-024-01616-7$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s12220-024-01616-7$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Domitrz, Wojciech</creatorcontrib><creatorcontrib>Hernández-Martínez, Lucía Ivonne</creatorcontrib><creatorcontrib>Sánchez-Bringas, Federico</creatorcontrib><title>On Singularities of the Gauss Map Components of Surfaces in R4</title><title>The Journal of geometric analysis</title><addtitle>J Geom Anal</addtitle><description>The Gauss map of a generic immersion of a smooth, oriented surface into
R
4
is an immersion. But this map takes values on the Grassmanian of oriented 2-planes in
R
4
. Since this manifold has a structure of a product of two spheres, the Gauss map has two components that take values on the sphere. We study the singularities of the components of the Gauss map and relate them to the geometric properties of the generic immersion. Moreover, we prove that the singularities are generically stable, and we connect them to the contact type of the surface and
J
-holomorphic curves with respect to an orthogonal complex structure
J
on
R
4
. Finally, we get some formulas of Gauss–Bonnet type involving the geometry of the singularities of the components with the geometry and topology of the surface.</description><subject>Abstract Harmonic Analysis</subject><subject>Convex and Discrete Geometry</subject><subject>Differential Geometry</subject><subject>Dynamical Systems and Ergodic Theory</subject><subject>Fourier Analysis</subject><subject>Global Analysis and Analysis on Manifolds</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Singularity (mathematics)</subject><subject>Submerging</subject><subject>Topology</subject><issn>1050-6926</issn><issn>1559-002X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNpFkE1LxDAQhoMouK7-AU8Bz9HJ5Ku9CFJ0FVYWXAVvIW1S7bK2tWn_v3EreJqBeZiZ9yHkksM1BzA3kSMiMEDJgGuumTkiC65UzgDw_Tj1oIDpHPUpOYtxByC1kGZBbjct3Tbtx7R3QzM2IdKupuNnoCs3xUifXU-L7qvv2tCOh9l2GmpXJa5p6Ys8Jye128dw8VeX5O3h_rV4ZOvN6qm4W7Me0YwMs1B5kRuOPOTGSWVMiS64ElD4rA4-A5WjV7XOQgZl-jkHqMA756XzQogluZr39kP3PYU42l03DW06aQUIxYWWQiZKzFTsh5QpDP8UB_srys6ibBJlD6KsET91Ulm9</recordid><startdate>20240601</startdate><enddate>20240601</enddate><creator>Domitrz, Wojciech</creator><creator>Hernández-Martínez, Lucía Ivonne</creator><creator>Sánchez-Bringas, Federico</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>C6C</scope><orcidid>https://orcid.org/0000-0003-3979-7101</orcidid></search><sort><creationdate>20240601</creationdate><title>On Singularities of the Gauss Map Components of Surfaces in R4</title><author>Domitrz, Wojciech ; Hernández-Martínez, Lucía Ivonne ; Sánchez-Bringas, Federico</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-p227t-28ecd397121e97a4577b2aeab023d8fed80592d5f68e80b002900c0daad4ad333</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Abstract Harmonic Analysis</topic><topic>Convex and Discrete Geometry</topic><topic>Differential Geometry</topic><topic>Dynamical Systems and Ergodic Theory</topic><topic>Fourier Analysis</topic><topic>Global Analysis and Analysis on Manifolds</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Singularity (mathematics)</topic><topic>Submerging</topic><topic>Topology</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Domitrz, Wojciech</creatorcontrib><creatorcontrib>Hernández-Martínez, Lucía Ivonne</creatorcontrib><creatorcontrib>Sánchez-Bringas, Federico</creatorcontrib><collection>Springer Nature OA Free Journals</collection><jtitle>The Journal of geometric analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Domitrz, Wojciech</au><au>Hernández-Martínez, Lucía Ivonne</au><au>Sánchez-Bringas, Federico</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On Singularities of the Gauss Map Components of Surfaces in R4</atitle><jtitle>The Journal of geometric analysis</jtitle><stitle>J Geom Anal</stitle><date>2024-06-01</date><risdate>2024</risdate><volume>34</volume><issue>6</issue><issn>1050-6926</issn><eissn>1559-002X</eissn><abstract>The Gauss map of a generic immersion of a smooth, oriented surface into
R
4
is an immersion. But this map takes values on the Grassmanian of oriented 2-planes in
R
4
. Since this manifold has a structure of a product of two spheres, the Gauss map has two components that take values on the sphere. We study the singularities of the components of the Gauss map and relate them to the geometric properties of the generic immersion. Moreover, we prove that the singularities are generically stable, and we connect them to the contact type of the surface and
J
-holomorphic curves with respect to an orthogonal complex structure
J
on
R
4
. Finally, we get some formulas of Gauss–Bonnet type involving the geometry of the singularities of the components with the geometry and topology of the surface.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s12220-024-01616-7</doi><orcidid>https://orcid.org/0000-0003-3979-7101</orcidid><oa>free_for_read</oa></addata></record> |
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| subjects | Abstract Harmonic Analysis Convex and Discrete Geometry Differential Geometry Dynamical Systems and Ergodic Theory Fourier Analysis Global Analysis and Analysis on Manifolds Mathematics Mathematics and Statistics Singularity (mathematics) Submerging Topology |
| title | On Singularities of the Gauss Map Components of Surfaces in R4 |
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