Differential Geometry and its Applications最新文献

{"title":"Classification of conformally flat Moebius isoparametric submanifolds in the Euclidean space","authors":"M.S.R. Antas","doi":"10.1016/j.difgeo.2024.102201","DOIUrl":"10.1016/j.difgeo.2024.102201","url":null,"abstract":"<div><div>The aim of this article is to classify umbilic-free isometric immersions <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>, of a conformally flat manifold which are Moebius isoparametric.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102201"},"PeriodicalIF":0.6,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660369","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":"Left-invariant pseudo-Riemannian metrics on Lie groups: The null cone","authors":"Sigbjørn Hervik","doi":"10.1016/j.difgeo.2024.102205","DOIUrl":"10.1016/j.difgeo.2024.102205","url":null,"abstract":"<div><div>We study left-invariant pseudo-Riemannian metrics on Lie groups using the moving bracket approach of the corresponding Lie algebra. We focus on metrics where the Lie algebra is in the null cone of the <span><math><mi>G</mi><mo>=</mo><mi>O</mi><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-action; i.e., Lie algebras <em>μ</em> where zero is in the closure of the orbits: <span><math><mn>0</mn><mo>∈</mo><mover><mrow><mi>G</mi><mo>⋅</mo><mi>μ</mi></mrow><mo>‾</mo></mover></math></span>. We provide examples of such Lie groups in various signatures and give some general results. For signatures <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> and <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> we classify all cases belonging to the null cone. More generally, we show that all nilpotent and completely solvable Lie algebras are in the null cone of some <span><math><mi>O</mi><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> action. In addition, several examples of non-trivial Levi-decomposable Lie algebras in the null cone are given.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102205"},"PeriodicalIF":0.6,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660362","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":"Singularities of discrete indefinite affine minimal surfaces","authors":"Marcos Craizer","doi":"10.1016/j.difgeo.2024.102206","DOIUrl":"10.1016/j.difgeo.2024.102206","url":null,"abstract":"<div><div>A smooth affine minimal surface with indefinite metric can be obtained from a pair of smooth non-intersecting spatial curves by Lelieuvre's formulas. These surfaces may present singularities, which are generically cuspidal edges and swallowtails. By discretizing the initial curves, one can obtain by the discrete Lelieuvre's formulas a discrete affine minimal surface with indefinite metric. The aim of this paper is to define the singular edges and vertices of the corresponding discrete asymptotic net in such a way that the most relevant properties of the singular set of the smooth version remain valid.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102206"},"PeriodicalIF":0.6,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660371","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":"Mean curvature flows of graphs sliding off to infinity in warped product manifolds","authors":"Naotoshi Fujihara","doi":"10.1016/j.difgeo.2024.102207","DOIUrl":"10.1016/j.difgeo.2024.102207","url":null,"abstract":"<div><div>We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and <span><math><mi>R</mi></math></span>. In such a warped product manifold, we can define the notion of a graph, called a geodesic graph. We prove that the curve shortening flow preserves a geodesic graph for any warping function, and the mean curvature flow of hypersurfaces preserves a geodesic graph for some monotone convex warping functions. In particular, we consider some warping functions that go to zero at infinity, which means that the curves or hypersurfaces go to a point at infinity along the flow. In such a case, we prove the long-time existence of the flow and that the curvature and its higher-order derivatives go to zero along the flow.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102207"},"PeriodicalIF":0.6,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660370","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":"On geodesics in the spaces of constrained curves","authors":"Esfandiar Nava-Yazdani","doi":"10.1016/j.difgeo.2024.102209","DOIUrl":"10.1016/j.difgeo.2024.102209","url":null,"abstract":"<div><div>In this work, we study the geodesics of the space of certain geometrically and physically motivated subspaces of the space of immersed curves endowed with a first order Sobolev metric. This includes elastic curves and also an extension of some results on planar concentric circles to surfaces. The work focuses on intrinsic and constructive approaches.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102209"},"PeriodicalIF":0.6,"publicationDate":"2024-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660373","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":"Generalized almost-Kähler–Ricci solitons","authors":"Michael Albanese ,&nbsp;Giuseppe Barbaro ,&nbsp;Mehdi Lejmi","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":"97 ","pages":"Article 102193"},"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":"Laiyuan Gao ,&nbsp;Horst Martini ,&nbsp;Deyan Zhang","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":"97 ","pages":"Article 102192"},"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":"Ognjen Milatovic","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":"97 ","pages":"Article 102191"},"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":"Xinlei Wu,&nbsp;Yanyan Sheng,&nbsp;Liang Zhang","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":"97 ","pages":"Article 102190"},"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":"Ragini Singhal","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":"97 ","pages":"Article 102187"},"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}