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Vector bundle automorphisms preserving Morse-Bott foliations 保持莫尔斯-波特叶形的矢量束自形变
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-09-12 DOI: 10.1016/j.difgeo.2024.102189
Sergiy Maksymenko
{"title":"Vector bundle automorphisms preserving Morse-Bott foliations","authors":"Sergiy Maksymenko","doi":"10.1016/j.difgeo.2024.102189","DOIUrl":"10.1016/j.difgeo.2024.102189","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":0.6,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142173424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a result of K. Okumura 关于 K. 奥村的一项成果
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-09-11 DOI: 10.1016/j.difgeo.2024.102188
Patrick J. Ryan
{"title":"On a result of K. Okumura","authors":"Patrick J. Ryan","doi":"10.1016/j.difgeo.2024.102188","DOIUrl":"10.1016/j.difgeo.2024.102188","url":null,"abstract":"<div><p>The purpose of this paper is to clarify and extend the result of K. Okumura in <span><span>[7]</span></span> concerning hypersurfaces in the non-flat complex space forms <span><math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> and <span><math><mi>C</mi><msup><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> whose *-Ricci tensor is <span><math><mi>D</mi></math></span>-recurrent.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102188"},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0926224524000810/pdfft?md5=ad2177aec7e5fc15bfcc3be1b916d84f&pid=1-s2.0-S0926224524000810-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142167492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Time-optimal solutions of Zermelo's navigation problem with moving obstacles 有移动障碍物的泽梅洛导航问题的时间最优解
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-09-09 DOI: 10.1016/j.difgeo.2024.102177
Zohreh Fathi , Behroz Bidabad
{"title":"Time-optimal solutions of Zermelo's navigation problem with moving obstacles","authors":"Zohreh Fathi ,&nbsp;Behroz Bidabad","doi":"10.1016/j.difgeo.2024.102177","DOIUrl":"10.1016/j.difgeo.2024.102177","url":null,"abstract":"<div><p>In this article, we study the Zermelo navigation problem with and without obstacles from a theoretical point of view and look towards some computational aspects. More intuitively, this navigation model is in fact an optimal control problem with continuous inequality constraints. We first aim to study the structure of these optimal trajectories using the geometric aspects of the problem. More precisely, we find the time-optimal trajectories and characterize them as geodesics of Randers metrics away from the danger zone and geodesics of (not necessarily Randers) Finsler metrics where they touch the boundary of the danger zone. We demonstrate some of the important behavior of these trajectories by examples. In particular, we will calculate these trajectories precisely for the critical case of an infinitesimal homothety which, in the language of optimal control problems, will be referred to in this paper as a <em>weak linear vortex</em>.</p><p>Regarding the computational aspects of the resulting optimal control problem with constraints and inspired by the geometry behind this problem, we propose a modification of the optimization scheme previously considered in [Li-Xu-Teo-Chu, Time-optimal Zermelo's navigation problem with moving and fixed obstacles, 2013] by adding a piecewise constant rotation. This modification will entail adding another piecewise constant control to the problem which in turn proves to make the resulting approximated time-optimal paths more precise and efficient as we argue by the example of navigation through a linear vortex.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102177"},"PeriodicalIF":0.6,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142158280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some results on Kenmotsu and Sasakian statistical manifolds 关于 Kenmotsu 和 Sasakian 统计流形的一些结果
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-09-09 DOI: 10.1016/j.difgeo.2024.102179
Fereshteh Malek, Parvin Fazlollahi
{"title":"Some results on Kenmotsu and Sasakian statistical manifolds","authors":"Fereshteh Malek,&nbsp;Parvin Fazlollahi","doi":"10.1016/j.difgeo.2024.102179","DOIUrl":"10.1016/j.difgeo.2024.102179","url":null,"abstract":"<div><p>In this paper, we mainly prove that on Kenmotsu and Sasakian statistical manifolds, the Riemannian curvature tensor and the statistical curvature tensor fields are equal, only if their covariant derivatives are equal.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102179"},"PeriodicalIF":0.6,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142164020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Multi-Dirac structures for Lie bialgebroids Lie 双桥体的多迪拉克结构
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-09-06 DOI: 10.1016/j.difgeo.2024.102178
Won-Hak Ri , Ju-Song Jong , Un-Gyong Jong , Kwang-Hyon Jong
{"title":"Multi-Dirac structures for Lie bialgebroids","authors":"Won-Hak Ri ,&nbsp;Ju-Song Jong ,&nbsp;Un-Gyong Jong ,&nbsp;Kwang-Hyon Jong","doi":"10.1016/j.difgeo.2024.102178","DOIUrl":"10.1016/j.difgeo.2024.102178","url":null,"abstract":"<div><p>In this paper, we introduce multi-Dirac structures for Lie bialgebroids, which generalize the multi-Dirac structures on manifolds and Dirac structures on Lie bialgebroids. Next, we also introduce higher-order Courant algebroids for Lie algebroids and higher-order Dorfman algebroids for Lie algebroids and study the relationship between them. Furthermore, we show that there is a one-to-one correspondence between the multi-Dirac structures for special Lie bialgebroids and the higher Dirac structures for Lie algebroids. Finally, we construct the Gerstenhaber algebra by using the multi-Dirac structure for Lie bialgebroids.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102178"},"PeriodicalIF":0.6,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142148095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Actions with cohomogeneity zero or one on the de Sitter space dSn−1,1 德西特空间 dSn-1,1 上同调为零或一的行为
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-09-03 DOI: 10.1016/j.difgeo.2024.102180
H. Mahdiloo , P. Ahmadi , M. Hassani
{"title":"Actions with cohomogeneity zero or one on the de Sitter space dSn−1,1","authors":"H. Mahdiloo ,&nbsp;P. Ahmadi ,&nbsp;M. Hassani","doi":"10.1016/j.difgeo.2024.102180","DOIUrl":"10.1016/j.difgeo.2024.102180","url":null,"abstract":"<div><p>The aim of this paper is to classify the connected Lie groups which act isometrically and with cohomogeneity <em>c</em>, where <span><math><mi>c</mi><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>, on the de Sitter space <span><math><mi>d</mi><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span> up to conjugacy in <span><math><mi>S</mi><mi>O</mi><mo>(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span> and then up to orbit equivalence. Among other results, we give the list of the groups represented in the isometry group of the de Sitter space <span><math><mi>d</mi><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102180"},"PeriodicalIF":0.6,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142128239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Modal fracture of higher groups 高等组的模态断裂
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-08-20 DOI: 10.1016/j.difgeo.2024.102176
David Jaz Myers
{"title":"Modal fracture of higher groups","authors":"David Jaz Myers","doi":"10.1016/j.difgeo.2024.102176","DOIUrl":"10.1016/j.difgeo.2024.102176","url":null,"abstract":"<div><p>In this paper, we examine the modal aspects of higher groups in Shulman's Cohesive Homotopy Type Theory. We show that every higher group sits within a modal fracture hexagon which renders it into its discrete, infinitesimal, and contractible components. This gives an unstable and synthetic construction of Schreiber's differential cohomology hexagon. As an example of this modal fracture hexagon, we recover the character diagram characterizing ordinary differential cohomology by its relation to its underlying integral cohomology and differential form data, although there is a subtle obstruction to generalizing the usual hexagon to higher types. Assuming the existence of a long exact sequence of differential form classifiers, we construct the classifiers for circle <em>k</em>-gerbes with connection and describe their modal fracture hexagon.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102176"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142011821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The volume of conformally flat manifolds as hypersurfaces in the light-cone 光锥中作为超曲面的保角平流形的体积
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-08-14 DOI: 10.1016/j.difgeo.2024.102173
Riku Kishida
{"title":"The volume of conformally flat manifolds as hypersurfaces in the light-cone","authors":"Riku Kishida","doi":"10.1016/j.difgeo.2024.102173","DOIUrl":"10.1016/j.difgeo.2024.102173","url":null,"abstract":"<div><p>In this paper, we focus on a conformally flat Riemannian manifold <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>)</mo></math></span> of dimension <em>n</em> isometrically immersed into the <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-dimensional light-cone <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> as a hypersurface. We compute the first and the second variational formulas on the volume of such hypersurfaces. Such a hypersurface <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is not only immersed in <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> but also isometrically realized as a hypersurface of a certain null hypersurface <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> in the Minkowski spacetime, which is different from <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>. Moreover, <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> has a volume-maximizing property in <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102173"},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141984634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A non-Vaisman LCK solvmanifold associated to a one-dimensional extension of a 2-step nilmanifold 与二阶零芒形的一维扩展相关的非瓦伊斯曼LCK求解芒形
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-08-12 DOI: 10.1016/j.difgeo.2024.102174
Hiroshi Sawai
{"title":"A non-Vaisman LCK solvmanifold associated to a one-dimensional extension of a 2-step nilmanifold","authors":"Hiroshi Sawai","doi":"10.1016/j.difgeo.2024.102174","DOIUrl":"10.1016/j.difgeo.2024.102174","url":null,"abstract":"<div><p>The purpose of this paper is to determine a locally conformal Kähler solvmanifold such that its associated solvable Lie group is a one-dimensional extension of a 2-step nilpotent Lie group.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102174"},"PeriodicalIF":0.6,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141964440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Singularities of focal sets of pseudo-spherical framed immersions in the three-dimensional anti-de Sitter space 三维反德西特空间中伪球面框架沉浸的焦点集奇点
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-08-08 DOI: 10.1016/j.difgeo.2024.102175
O. Oğulcan Tuncer
{"title":"Singularities of focal sets of pseudo-spherical framed immersions in the three-dimensional anti-de Sitter space","authors":"O. Oğulcan Tuncer","doi":"10.1016/j.difgeo.2024.102175","DOIUrl":"10.1016/j.difgeo.2024.102175","url":null,"abstract":"<div><p>We introduce pseudo-spherical non-null framed curves in the three-dimensional anti-de Sitter spacetime and establish the existence and uniqueness of these curves. We then give moving frames along pseudo-spherical framed curves, which are well-defined even at singular points of the curve. These moving frames enable us to define evolutes and focal surfaces of pseudo-spherical framed immersions. We investigate the singularity properties of these evolutes and focal surfaces. We then reveal that the evolute of a pseudo-spherical framed immersion is the set of singular points of its focal surface. We also interpret evolutes and focal surfaces as the discriminant and the secondary discriminant sets of certain height functions, which allows us to explain evolutes and focal surfaces as wavefronts from the viewpoint of Legendrian singularity theory. Examples are provided to flesh out our results, and we use the hyperbolic Hopf map to visualize these examples.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102175"},"PeriodicalIF":0.6,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141950695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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