The Global Property of Generic Conformally Flat Hypersurfaces in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula>
A conformally flat hypersurface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>M</mi><mn>3</mn></msup><mo>→&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/6/1435 |
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| author | Yayun Chen Tongzhu Li |
| author_facet | Yayun Chen Tongzhu Li |
| author_sort | Yayun Chen |
| collection | DOAJ |
| description | A conformally flat hypersurface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>M</mi><mn>3</mn></msup><mo>→</mo><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></mrow></semantics></math></inline-formula> in the four-dimensional Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula> is said to be generic if the hypersurface has three distinct principal curvatures everywhere. In this paper, we study the generic conformally flat hypersurfaces in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula> using the framework of Möbius geometry. First, we classify locally the generic conformally flat hypersurfaces with a vanishing Möbius form under the Möbius transformation group of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula>. Second, we investigate the global behavior of the compact generic conformally flat hypersurfaces and give some integral formulas about the Möbius invariant of these hypersurfaces. |
| first_indexed | 2024-03-11T06:13:02Z |
| format | Article |
| id | doaj.art-17c2d35697574800943351b036dd003d |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T06:13:02Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-17c2d35697574800943351b036dd003d2023-11-17T12:28:34ZengMDPI AGMathematics2227-73902023-03-01116143510.3390/math11061435The Global Property of Generic Conformally Flat Hypersurfaces in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula>Yayun Chen0Tongzhu Li1Department of Mathematics, Beijing Institute of Technology, Beijing 100081, ChinaDepartment of Mathematics, Beijing Institute of Technology, Beijing 100081, ChinaA conformally flat hypersurface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>M</mi><mn>3</mn></msup><mo>→</mo><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></mrow></semantics></math></inline-formula> in the four-dimensional Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula> is said to be generic if the hypersurface has three distinct principal curvatures everywhere. In this paper, we study the generic conformally flat hypersurfaces in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula> using the framework of Möbius geometry. First, we classify locally the generic conformally flat hypersurfaces with a vanishing Möbius form under the Möbius transformation group of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula>. Second, we investigate the global behavior of the compact generic conformally flat hypersurfaces and give some integral formulas about the Möbius invariant of these hypersurfaces.https://www.mdpi.com/2227-7390/11/6/1435generic conformally flat hypersurfaceMöbius metricMöbius formMöbius curvature |
| spellingShingle | Yayun Chen Tongzhu Li The Global Property of Generic Conformally Flat Hypersurfaces in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula> Mathematics generic conformally flat hypersurface Möbius metric Möbius form Möbius curvature |
| title | The Global Property of Generic Conformally Flat Hypersurfaces in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula> |
| title_full | The Global Property of Generic Conformally Flat Hypersurfaces in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula> |
| title_fullStr | The Global Property of Generic Conformally Flat Hypersurfaces in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula> |
| title_full_unstemmed | The Global Property of Generic Conformally Flat Hypersurfaces in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula> |
| title_short | The Global Property of Generic Conformally Flat Hypersurfaces in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>4</mn></msup></semantics></math></inline-formula> |
| title_sort | global property of generic conformally flat hypersurfaces in inline formula math display inline semantics msup mi mathvariant double struck r mi mn 4 mn msup semantics math inline formula |
| topic | generic conformally flat hypersurface Möbius metric Möbius form Möbius curvature |
| url | https://www.mdpi.com/2227-7390/11/6/1435 |
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