{"title":"R4 中球面的奇异点","authors":"Haiming Liu, Yuefeng Hua, Wanzhen Li","doi":"10.1515/math-2024-0033","DOIUrl":null,"url":null,"abstract":"In this article, we mainly study the geometric properties of spherical surface of a curve on a hypersurface <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0033_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Σ</m:mi> </m:math> <jats:tex-math>\\Sigma </jats:tex-math> </jats:alternatives> </jats:inline-formula> in four-dimensional Euclidean space. We define a family of tangent height functions of a curve on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0033_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Σ</m:mi> </m:math> <jats:tex-math>\\Sigma </jats:tex-math> </jats:alternatives> </jats:inline-formula> as the main tool for research and combine the relevant knowledge of singularity theory. It is shown that there are three types of singularities of spherical surface, that is, in the local sense, the spherical surface is respectively diffeomorphic to the cuspidal edge, the swallowtail, and the cuspidal beaks. In addition, we give two examples of the spherical surface.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"25 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Singularities of spherical surface in R4\",\"authors\":\"Haiming Liu, Yuefeng Hua, Wanzhen Li\",\"doi\":\"10.1515/math-2024-0033\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this article, we mainly study the geometric properties of spherical surface of a curve on a hypersurface <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2024-0033_eq_001.png\\\"/> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"normal\\\">Σ</m:mi> </m:math> <jats:tex-math>\\\\Sigma </jats:tex-math> </jats:alternatives> </jats:inline-formula> in four-dimensional Euclidean space. We define a family of tangent height functions of a curve on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2024-0033_eq_002.png\\\"/> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"normal\\\">Σ</m:mi> </m:math> <jats:tex-math>\\\\Sigma </jats:tex-math> </jats:alternatives> </jats:inline-formula> as the main tool for research and combine the relevant knowledge of singularity theory. It is shown that there are three types of singularities of spherical surface, that is, in the local sense, the spherical surface is respectively diffeomorphic to the cuspidal edge, the swallowtail, and the cuspidal beaks. In addition, we give two examples of the spherical surface.\",\"PeriodicalId\":48713,\"journal\":{\"name\":\"Open Mathematics\",\"volume\":\"25 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-08-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Open Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/math-2024-0033\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Open Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/math-2024-0033","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
In this article, we mainly study the geometric properties of spherical surface of a curve on a hypersurface Σ\Sigma in four-dimensional Euclidean space. We define a family of tangent height functions of a curve on Σ\Sigma as the main tool for research and combine the relevant knowledge of singularity theory. It is shown that there are three types of singularities of spherical surface, that is, in the local sense, the spherical surface is respectively diffeomorphic to the cuspidal edge, the swallowtail, and the cuspidal beaks. In addition, we give two examples of the spherical surface.
期刊介绍:
Open Mathematics - formerly Central European Journal of Mathematics
Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant, original and relevant works in all areas of mathematics. The journal provides the readers with free, instant, and permanent access to all content worldwide; and the authors with extensive promotion of published articles, long-time preservation, language-correction services, no space constraints and immediate publication.
Open Mathematics is listed in Thomson Reuters - Current Contents/Physical, Chemical and Earth Sciences. Our standard policy requires each paper to be reviewed by at least two Referees and the peer-review process is single-blind.
Aims and Scope
The journal aims at presenting high-impact and relevant research on topics across the full span of mathematics. Coverage includes: