Differential Geometry and its Applications最新文献

{"title":"Generalized almost-Kähler–Ricci solitons","authors":"","doi":"10.1016/j.difgeo.2024.102193","DOIUrl":"10.1016/j.difgeo.2024.102193","url":null,"abstract":"<div><div>We generalize Kähler–Ricci solitons to the almost-Kähler setting as the zeros of Inoue's moment map <span><span>[25]</span></span>, and show that their existence is an obstruction to the existence of first-Chern–Einstein almost-Kähler metrics on compact symplectic Fano manifolds. We prove deformation results of such metrics in the 4-dimensional case. Moreover, we study the Lie algebra of holomorphic vector fields on 2<em>n</em>-dimensional compact symplectic Fano manifolds admitting generalized almost-Kähler–Ricci solitons. In particular, we partially extend Matsushima's theorem <span><span>[41]</span></span> to compact first-Chern–Einstein almost-Kähler manifolds.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533161","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":"Deforming locally convex curves into curves of constant k-order width","authors":"","doi":"10.1016/j.difgeo.2024.102192","DOIUrl":"10.1016/j.difgeo.2024.102192","url":null,"abstract":"<div><div>A nonlocal curvature flow is introduced to evolve locally convex curves in the plane. It is proved that this flow with any initial locally convex curve has a global solution, keeping the local convexity and the elastic energy of the evolving curve, and that, as the time goes to infinity, the curve converges to a smooth, locally convex curve of constant <em>k</em>-order width. In particular, the limiting curve is a multiple circle if and only if the initial locally convex curve is <em>k</em>-symmetric.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533160","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":"Covariant Schrödinger operator and L2-vanishing property on Riemannian manifolds","authors":"","doi":"10.1016/j.difgeo.2024.102191","DOIUrl":"10.1016/j.difgeo.2024.102191","url":null,"abstract":"<div><p>Let <em>M</em> be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let <span><math><mi>E</mi></math></span> be a Hermitian vector bundle over <em>M</em> equipped with a metric covariant derivative ∇. We consider the operator <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mo>,</mo><mi>V</mi></mrow></msub><mo>=</mo><msup><mrow><mi>∇</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>∇</mi><mo>+</mo><msub><mrow><mi>∇</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>+</mo><mi>V</mi></math></span>, where <span><math><msup><mrow><mi>∇</mi></mrow><mrow><mi>†</mi></mrow></msup></math></span> is the formal adjoint of ∇ with respect to the inner product in the space of square-integrable sections of <span><math><mi>E</mi></math></span>, <em>X</em> is a smooth (real) vector field on <em>M</em>, and <em>V</em> is a fiberwise self-adjoint, smooth section of the endomorphism bundle <span><math><mi>End</mi><mspace></mspace><mi>E</mi></math></span>. We give a sufficient condition for the triviality of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-kernel of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mo>,</mo><mi>V</mi></mrow></msub></math></span>. As a corollary, putting <span><math><mi>X</mi><mo>≡</mo><mn>0</mn></math></span> and working in the setting of a Clifford module equipped with a Clifford connection ∇, we obtain the triviality of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-kernel of <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, where <em>D</em> is the Dirac operator corresponding to ∇. In particular, when <span><math><mi>E</mi><mo>=</mo><msubsup><mrow><mi>Λ</mi></mrow><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msubsup><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>M</mi></math></span> and <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is the Hodge–deRham Laplacian on (complex-valued) <em>k</em>-forms, we recover some recent vanishing results for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-harmonic (complex-valued) <em>k</em>-forms.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142228643","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 Sasakian statistical structures of constant ϕ-sectional curvature on Sasakian space forms","authors":"","doi":"10.1016/j.difgeo.2024.102190","DOIUrl":"10.1016/j.difgeo.2024.102190","url":null,"abstract":"<div><p>In this paper, we investigate the Sasakian statistical structures of constant <em>ϕ</em>-sectional curvature based on Sasakian space forms. We obtain the classification of this kind of Sasakian statistical structures. Our classification results show that the Sasakian statistical structures of constant <em>ϕ</em>-sectional curvature on a Sasakian space form with dimension higher than 3 must be almost-trivial; on a 3-dimensional Sasakian space form, in addition to the almost-trivial Sasakian statistical structure, there exist other Sasakian statistical structures which satisfy the constant <em>ϕ</em>-sectional curvature condition. We also point out that a rigidity result for cosymplectic statistical structures of constant <em>ϕ</em>-sectional curvature on 3-dimensional cosymplectic space forms in <span><span>[11]</span></span> can be improved to the corresponding classification result.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142173432","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":"Nearly half-flat SU(3) structures on S3 × S3","authors":"","doi":"10.1016/j.difgeo.2024.102187","DOIUrl":"10.1016/j.difgeo.2024.102187","url":null,"abstract":"<div><p>We study the <span><math><mi>SU</mi><mo>(</mo><mn>3</mn><mo>)</mo></math></span>-structure induced on an oriented hypersurface of a 7-dimensional manifold with a nearly parallel <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-structure. Such <span><math><mi>SU</mi><mo>(</mo><mn>3</mn><mo>)</mo></math></span>-structures are called <em>nearly half-flat</em>. We characterise the left invariant nearly half-flat structures on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. This characterisation then helps us to systematically analyse nearly parallel <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-structures on an interval times <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142173423","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":"Vector bundle automorphisms preserving Morse-Bott foliations","authors":"","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":null,"pages":null},"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}

{"title":"On a result of K. Okumura","authors":"","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":null,"pages":null},"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}

{"title":"Time-optimal solutions of Zermelo's navigation problem with moving obstacles","authors":"","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":null,"pages":null},"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":"","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":null,"pages":null},"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":"","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":null,"pages":null},"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}