{"title":"通过投影和法向截面联系流形的二阶几何","authors":"P. B. Riul, R. O. Sinha","doi":"10.5565/publmat6512114","DOIUrl":null,"url":null,"abstract":"We use normal sections to relate the curvature locus of regular (resp. singular corank 1) 3-manifolds in $\\mathbb{R}^6$ (resp. $\\mathbb R^5$) with regular (resp. singular corank 1) surfaces in $\\mathbb R^5$ (resp. $\\mathbb R^4$). For example we show how to generate a Roman surface by a family of ellipses different to Steiner's way. Furthermore, we give necessary conditions for the 2-jet of the parametrisation of a singular 3-manifold to be in a certain orbit in terms of the topological types of the curvature loci of the singular surfaces obtained as normal sections. We also study the relations between the regular and singular cases through projections. We show there is a commutative diagram of projections and normal sections which relates the curvature loci of the different types of manifolds, and therefore, that the second order geometry of all of them is related. In particular we define asymptotic directions for singular corank 1 3-manifolds in $\\mathbb R^5$ and relate them to asymptotic directions of regular 3-manifolds in $\\mathbb R^6$ and singular corank 1 surfaces in $\\mathbb R^4$.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2019-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"Relating second order geometry of manifolds through projections and normal sections\",\"authors\":\"P. B. Riul, R. O. Sinha\",\"doi\":\"10.5565/publmat6512114\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We use normal sections to relate the curvature locus of regular (resp. singular corank 1) 3-manifolds in $\\\\mathbb{R}^6$ (resp. $\\\\mathbb R^5$) with regular (resp. singular corank 1) surfaces in $\\\\mathbb R^5$ (resp. $\\\\mathbb R^4$). For example we show how to generate a Roman surface by a family of ellipses different to Steiner's way. Furthermore, we give necessary conditions for the 2-jet of the parametrisation of a singular 3-manifold to be in a certain orbit in terms of the topological types of the curvature loci of the singular surfaces obtained as normal sections. We also study the relations between the regular and singular cases through projections. We show there is a commutative diagram of projections and normal sections which relates the curvature loci of the different types of manifolds, and therefore, that the second order geometry of all of them is related. In particular we define asymptotic directions for singular corank 1 3-manifolds in $\\\\mathbb R^5$ and relate them to asymptotic directions of regular 3-manifolds in $\\\\mathbb R^6$ and singular corank 1 surfaces in $\\\\mathbb R^4$.\",\"PeriodicalId\":54531,\"journal\":{\"name\":\"Publicacions Matematiques\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2019-09-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Publicacions Matematiques\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.5565/publmat6512114\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Publicacions Matematiques","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.5565/publmat6512114","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Relating second order geometry of manifolds through projections and normal sections
We use normal sections to relate the curvature locus of regular (resp. singular corank 1) 3-manifolds in $\mathbb{R}^6$ (resp. $\mathbb R^5$) with regular (resp. singular corank 1) surfaces in $\mathbb R^5$ (resp. $\mathbb R^4$). For example we show how to generate a Roman surface by a family of ellipses different to Steiner's way. Furthermore, we give necessary conditions for the 2-jet of the parametrisation of a singular 3-manifold to be in a certain orbit in terms of the topological types of the curvature loci of the singular surfaces obtained as normal sections. We also study the relations between the regular and singular cases through projections. We show there is a commutative diagram of projections and normal sections which relates the curvature loci of the different types of manifolds, and therefore, that the second order geometry of all of them is related. In particular we define asymptotic directions for singular corank 1 3-manifolds in $\mathbb R^5$ and relate them to asymptotic directions of regular 3-manifolds in $\mathbb R^6$ and singular corank 1 surfaces in $\mathbb R^4$.
期刊介绍:
Publicacions Matemàtiques is a research mathematical journal published by the Department of Mathematics of the Universitat Autònoma de Barcelona since 1976 (before 1988 named Publicacions de la Secció de Matemàtiques, ISSN: 0210-2978 print, 2014-4369 online). Two issues, constituting a single volume, are published each year. The journal has a large circulation being received by more than two hundred libraries all over the world. It is indexed by Mathematical Reviews, Zentralblatt Math., Science Citation Index, SciSearch®, ISI Alerting Services, COMPUMATH Citation Index®, and it participates in the Euclid Project and JSTOR. Free access is provided to all published papers through the web page.
Publicacions Matemàtiques is a non-profit university journal which gives special attention to the authors during the whole editorial process. In 2019, the average time between the reception of a paper and its publication was twenty-two months, and the average time between the acceptance of a paper and its publication was fifteen months. The journal keeps on receiving a large number of submissions, so the authors should be warned that currently only articles with excellent reports can be accepted.