{"title":"给定常秩锚映射的切束上李代数群的存在性","authors":"J. Monterde","doi":"10.1016/j.difgeo.2023.102040","DOIUrl":null,"url":null,"abstract":"<div><p>We show that given a constant rank linear map, <span><math><mi>K</mi><mo>:</mo><mi>T</mi><mi>M</mi><mo>→</mo><mi>T</mi><mi>M</mi></math></span>, there exists a Lie algebroid with <em>K</em> as its anchor map, if and only if the image distribution, Im<em>K</em>, is involutive. As a byproduct, a new example of Lie algebroid bracket associated with a regular foliation is obtained through the projector onto the involutive distribution. The Lie algebroid bracket is not defined on the involutive distribution but on the whole space of vector fields of the manifold.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence of Lie algebroids on the tangent bundle with a given anchor map of constant rank\",\"authors\":\"J. Monterde\",\"doi\":\"10.1016/j.difgeo.2023.102040\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We show that given a constant rank linear map, <span><math><mi>K</mi><mo>:</mo><mi>T</mi><mi>M</mi><mo>→</mo><mi>T</mi><mi>M</mi></math></span>, there exists a Lie algebroid with <em>K</em> as its anchor map, if and only if the image distribution, Im<em>K</em>, is involutive. As a byproduct, a new example of Lie algebroid bracket associated with a regular foliation is obtained through the projector onto the involutive distribution. The Lie algebroid bracket is not defined on the involutive distribution but on the whole space of vector fields of the manifold.</p></div>\",\"PeriodicalId\":51010,\"journal\":{\"name\":\"Differential Geometry and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Differential Geometry and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0926224523000669\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Geometry and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0926224523000669","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Existence of Lie algebroids on the tangent bundle with a given anchor map of constant rank
We show that given a constant rank linear map, , there exists a Lie algebroid with K as its anchor map, if and only if the image distribution, ImK, is involutive. As a byproduct, a new example of Lie algebroid bracket associated with a regular foliation is obtained through the projector onto the involutive distribution. The Lie algebroid bracket is not defined on the involutive distribution but on the whole space of vector fields of the manifold.
期刊介绍:
Differential Geometry and its Applications publishes original research papers and survey papers in differential geometry and in all interdisciplinary areas in mathematics which use differential geometric methods and investigate geometrical structures. The following main areas are covered: differential equations on manifolds, global analysis, Lie groups, local and global differential geometry, the calculus of variations on manifolds, topology of manifolds, and mathematical physics.