{"title":"Vector bundle automorphisms preserving Morse-Bott foliations","authors":"","doi":"10.1016/j.difgeo.2024.102189","DOIUrl":null,"url":null,"abstract":"<div><p>Let <em>M</em> be a smooth manifold and <span><math><mi>F</mi></math></span> a Morse-Bott foliation with a compact critical manifold <span><math><mi>Σ</mi><mo>⊂</mo><mi>M</mi></math></span>. Denote by <span><math><mi>D</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span> the group of diffeomorphisms of <em>M</em> leaving invariant each leaf of <span><math><mi>F</mi></math></span>. Under certain assumptions on <span><math><mi>F</mi></math></span> it is shown that the computation of the homotopy type of <span><math><mi>D</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span> reduces to three rather independent groups: the group of diffeomorphisms of Σ, the group of vector bundle automorphisms of some regular neighborhood of Σ, and the subgroup of <span><math><mi>D</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span> consisting of diffeomorphisms fixed near Σ. Examples of computations of homotopy types of groups <span><math><mi>D</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span> for such foliations are also presented.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-09-12","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/S0926224524000822","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let M be a smooth manifold and a Morse-Bott foliation with a compact critical manifold . Denote by the group of diffeomorphisms of M leaving invariant each leaf of . Under certain assumptions on it is shown that the computation of the homotopy type of reduces to three rather independent groups: the group of diffeomorphisms of Σ, the group of vector bundle automorphisms of some regular neighborhood of Σ, and the subgroup of consisting of diffeomorphisms fixed near Σ. Examples of computations of homotopy types of groups for such foliations are also presented.
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.