{"title":"保持莫尔斯-波特叶形的矢量束自形变","authors":"Sergiy Maksymenko","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":"97 ","pages":"Article 102189"},"PeriodicalIF":0.6000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Vector bundle automorphisms preserving Morse-Bott foliations\",\"authors\":\"Sergiy Maksymenko\",\"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\":\"97 \",\"pages\":\"Article 102189\"},\"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}","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
摘要
假设 M 是光滑流形,F 是莫尔斯-波特流形,且有一个紧凑临界流形 Σ⊂M。在 F 的某些假设条件下,D(F) 的同调类型的计算可以简化为三个独立的群:Σ 的差分变形群、Σ 的某个规则邻域的向量束自动变形群以及由固定在 Σ 附近的差分变形组成的 D(F) 子群。文中还举例说明了此类叶形的群 D(F) 的同调类型计算。
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.