Differential Geometry and its Applications最新文献

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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
{"title":"Actions with cohomogeneity zero or one on the de Sitter space dSn−1,1","authors":"","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":null,"pages":null},"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
{"title":"Modal fracture of higher groups","authors":"","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":null,"pages":null},"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
{"title":"The volume of conformally flat manifolds as hypersurfaces in the light-cone","authors":"","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":null,"pages":null},"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
{"title":"A non-Vaisman LCK solvmanifold associated to a one-dimensional extension of a 2-step nilmanifold","authors":"","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":null,"pages":null},"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
{"title":"Singularities of focal sets of pseudo-spherical framed immersions in the three-dimensional anti-de Sitter space","authors":"","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":null,"pages":null},"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
Existence and density results of conformal metrics with prescribed higher order Q-curvature on Sn Sn上具有规定高阶Q曲率的共形度量的存在性和密度结果
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-08-01 DOI: 10.1016/j.difgeo.2024.102172
{"title":"Existence and density results of conformal metrics with prescribed higher order Q-curvature on Sn","authors":"","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":null,"pages":null},"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}
引用次数: 0
Diffeological submanifolds and their friends 衍射子平面及其朋友们
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-08-01 DOI: 10.1016/j.difgeo.2024.102170
{"title":"Diffeological submanifolds and their friends","authors":"","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":null,"pages":null},"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}
引用次数: 0
Topology of toric gravitational instantons 环状引力瞬子拓扑学
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-07-31 DOI: 10.1016/j.difgeo.2024.102171
{"title":"Topology of toric gravitational instantons","authors":"","doi":"10.1016/j.difgeo.2024.102171","DOIUrl":"10.1016/j.difgeo.2024.102171","url":null,"abstract":"<div><p>For an asymptotically locally Euclidean (ALE) or asymptotically locally flat (ALF) gravitational instanton <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> with toric symmetry, we express the signature of <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> directly in terms of its rod structure. Applying Hitchin–Thorpe-type inequalities for Ricci-flat ALE/ALF manifolds, we formulate, as a step toward a classification of toric ALE/ALF instantons, necessary conditions that the rod structures of such spaces must satisfy. Finally, we apply these results to the study of rod structures with three turning points.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0926224524000640/pdfft?md5=1af94bc08a68f11151c59c10b99043ce&pid=1-s2.0-S0926224524000640-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141961512","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
Schwarz lemma for conformal parametrization of minimal graphs in M×R M×R 中最小图共形参数化的 Schwarz Lemma
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-06-26 DOI: 10.1016/j.difgeo.2024.102169
David Kalaj
{"title":"Schwarz lemma for conformal parametrization of minimal graphs in M×R","authors":"David Kalaj","doi":"10.1016/j.difgeo.2024.102169","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102169","url":null,"abstract":"<div><p>We prove Schwarz-type lemma results for Weierstrass parameterization of the minimal disk in the Riemannian manifold <span><math><mi>M</mi><mo>×</mo><mi>R</mi></math></span>, where <em>M</em> is a Riemannian surface of non-positive Gaussian curvature. The estimate is sharp, and the equality is attained if and only if the <em>ϱ</em>-harmonic mapping that produces the parameterization is conformal and the metric is a Euclidean metric. If the area of the minimal surface is equal to the area of the disk, then the parametrization is a contraction w.r.t. induced metric and hyperbolic metric respectively.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481516","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
Equivariant harmonic maps of the complex projective spaces into the quaternion projective spaces 复投影空间到四元投影空间的等调和映射
IF 0.6 4区 数学
Differential Geometry and its Applications Pub Date : 2024-06-26 DOI: 10.1016/j.difgeo.2024.102167
Isami Koga , Yasuyuki Nagatomo
{"title":"Equivariant harmonic maps of the complex projective spaces into the quaternion projective spaces","authors":"Isami Koga ,&nbsp;Yasuyuki Nagatomo","doi":"10.1016/j.difgeo.2024.102167","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102167","url":null,"abstract":"<div><p>We classify equivariant harmonic maps of the complex projective spaces <span><math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> into the quaternion projective spaces. To do this, we employ differential geometry of vector bundles and connections. When the domain is the complex projective <em>line</em>, we have one parameter family of those maps. (This result is already shown in <span>[2]</span> and <span>[4]</span> in other ways). However, when <span><math><mi>m</mi><mo>≧</mo><mn>2</mn></math></span>, we will obtain the rigidity results.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0926224524000603/pdfft?md5=50c3b21df49c5a546924763a29df2d65&pid=1-s2.0-S0926224524000603-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481519","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
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