Differential Geometry and its Applications最新文献

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"title":"Existence and density results of conformal metrics with prescribed higher order Q-curvature on Sn","authors":"Zhongwei Tang ,&nbsp;Heming Wang ,&nbsp;Ning Zhou","doi":"10.1016/j.difgeo.2024.102172","DOIUrl":"10.1016/j.difgeo.2024.102172","url":null,"abstract":"<div><p>We prove some results on the density and multiplicity of positive solutions to the conformal <em>Q</em>-curvature equations <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mi>K</mi><msup><mrow><mi>v</mi></mrow><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn><mi>m</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>m</mi></mrow></mfrac></mrow></msup></math></span> on the <em>n</em>-dimensional standard unit sphere <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> for all <span><math><mi>m</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></math></span> and <em>m</em> is an integer, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> is the intertwining operator of order 2<em>m</em> and <em>K</em> is the prescribed <em>Q</em>-curvature function. More specifically, by using the variational gluing method, refined analysis of bubbling behavior, Pohozaev identity, as well as the blow up argument for nonlinear integral equations, we construct arbitrarily many multi-bump solutions. In particular, we show the smooth positive <em>Q</em>-curvature functions of metrics conformal to <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> are dense in the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> topology. Existence results of infinitely many positive solutions to the poly-harmonic equations <span><math><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>m</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>K</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mi>u</mi></mrow><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn><mi>m</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>m</mi></mrow></mfrac></mrow></msup></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><mi>K</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> being asymptotically periodic are also obtained.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102172"},"PeriodicalIF":0.6,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141961513","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}

{"title":"Diffeological submanifolds and their friends","authors":"Yael Karshon ,&nbsp;David Miyamoto ,&nbsp;Jordan Watts","doi":"10.1016/j.difgeo.2024.102170","DOIUrl":"10.1016/j.difgeo.2024.102170","url":null,"abstract":"<div><p>A smooth manifold hosts different types of submanifolds, including embedded, weakly-embedded, and immersed submanifolds. The notion of an immersed submanifold requires additional structure (namely, the choice of a topology); when this additional structure is unique, we call the subset a <em>uniquely immersed submanifold</em>. Diffeology provides yet another intrinsic notion of submanifold: a <em>diffeological submanifold</em>.</p><p>We show that from a categorical perspective diffeology rises above the others: viewing manifolds as a concrete category over the category of sets, the <em>initial morphisms</em> are exactly the (diffeological) <em>inductions</em>, which are the diffeomorphisms with diffeological submanifolds. Moreover, if we view manifolds as a concrete category over the category of topological spaces, we recover Joris and Preissmann's notion of <em>pseudo-immersions</em>.</p><p>We show that these notions are all different. In particular, a theorem of Joris from 1982 yields a diffeological submanifold whose inclusion is not an immersion, answering a question that was posed by Iglesias-Zemmour. We also characterize local inductions as those pseudo-immersions that are locally injective.</p><p>In appendices, we review a proof of Joris' theorem, pointing at a flaw in one of the several other proofs that occur in the literature, and we illustrate how submanifolds inherit paracompactness from their ambient manifold.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102170"},"PeriodicalIF":0.6,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141951372","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}