Differential Geometry and its Applications最新文献

{"title":"Absolutely continuous curves in Finsler-like spaces","authors":"Fue Zhang ,&nbsp;Wei Zhao","doi":"10.1016/j.difgeo.2024.102154","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102154","url":null,"abstract":"<div><p>The present paper is devoted to the investigation of absolutely continuous curves in asymmetric metric spaces induced by Finsler structures. Firstly, for asymmetric spaces induced by Finsler manifolds, we show that three different kinds of absolutely continuous curves coincide when their domains are bounded closed intervals. As an application, a universal existence and regularity theorem for gradient flow is obtained in the Finsler setting. Secondly, we study absolutely continuous curves in Wasserstein spaces over Finsler manifolds and establish the Lisini structure theorem in this setting, which characterize the nature of absolutely continuous curves in Wasserstein spaces in terms of dynamical transference plans concentrated on absolutely continuous curves in base Finsler manifolds. Besides, a close relation between continuity equations and absolutely continuous curves in Wasserstein spaces is founded. Last but not least, we also consider nonsmooth “Finsler-like” spaces, in which case most of the aforementioned results remain valid. Various model examples are constructed in this paper, which point out genuine differences between the asymmetric and symmetric settings.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"96 ","pages":"Article 102154"},"PeriodicalIF":0.5,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141084325","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":"Almost-Kähler four-manifolds with harmonic self-dual Weyl curvature","authors":"Inyoung Kim","doi":"10.1016/j.difgeo.2024.102141","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102141","url":null,"abstract":"<div><p>We show that a compact almost-Kähler four-manifold <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>ω</mi><mo>)</mo></math></span> with harmonic self-dual Weyl curvature and constant scalar curvature is Kähler if <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⋅</mo><mo>[</mo><mi>ω</mi><mo>]</mo><mo>≥</mo><mn>0</mn></math></span>. We also prove an integral curvature inequality for compact almost-Kähler four-manifolds with harmonic self-dual Weyl curvature.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"94 ","pages":"Article 102141"},"PeriodicalIF":0.5,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140813458","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 the splitting of weak nearly cosymplectic manifolds","authors":"Vladimir Rovenski","doi":"10.1016/j.difgeo.2024.102142","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102142","url":null,"abstract":"<div><p>Weak almost contact metric manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak (2022), allowed a new look at the theory of contact manifolds. This paper studies the curvature and topology of new structures of this type, called the weak nearly cosymplectic structure and weak nearly Kähler structure. We find conditions under which weak nearly cosymplectic manifolds become Riemannian products and characterize 5-dimensional weak nearly cosymplectic manifolds. Our theorems generalize results by H. Endo (2005) and A. Nicola–G. Dileo–I. Yudin (2018) to the context of weak almost contact geometry.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"94 ","pages":"Article 102142"},"PeriodicalIF":0.5,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140807470","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 stability of subelliptic harmonic maps with potential","authors":"Tian Chong ,&nbsp;Yuxin Dong ,&nbsp;Guilin Yang","doi":"10.1016/j.difgeo.2024.102143","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102143","url":null,"abstract":"<div><p>In this paper, we investigate the stability problem of subelliptic harmonic maps with potential. First, we derive the first and second variation formulas for subelliptic harmonic maps with potential. As a result, it is proved that a subelliptic harmonic map with potential is stable if the target manifold has nonpositive curvature and the Hessian of the potential is nonpositive definite. We also give Leung type results which involve the instability of subelliptic harmonic maps with potential when the target manifold is a sphere of dimension ≥3.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"94 ","pages":"Article 102143"},"PeriodicalIF":0.5,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140643595","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":"Transversality of the perturbed reduced Vafa-Witten moduli spaces on 4-manifolds","authors":"Ren Guan","doi":"10.1016/j.difgeo.2024.102139","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102139","url":null,"abstract":"<div><p>Previously we finish the establishment of the transversality of the general part of the Vafa-Witten moduli spaces, in this paper, we deal with the rest, i.e., the reduced part. We consider Vafa-Witten equation on closed, oriented and smooth Riemann 4-manifolds with <span><math><mi>C</mi><mo>≡</mo><mn>0</mn></math></span>, and construct perturbation to establish the transversality of the perturbed equation. We show that for a generic choice of the perturbation terms, the moduli space of solutions to the perturbed reduced Vafa-Witten equation for the structure group <span><math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math></span> or <span><math><mi>S</mi><mi>O</mi><mo>(</mo><mn>3</mn><mo>)</mo></math></span> on a closed 4-manifold is a smooth manifold of dimension zero. Finally we prove that for two generic orientation-preserving parameters, the corresponding moduli spaces are cobordant, and the method can also be applied to the general part.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"94 ","pages":"Article 102139"},"PeriodicalIF":0.5,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140618045","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":"Characterization of invariant complex Finsler metrics on the complex Grassmann manifold","authors":"Pandeng Cao,&nbsp;Xiaoshu Ge,&nbsp;Chunping Zhong","doi":"10.1016/j.difgeo.2024.102138","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102138","url":null,"abstract":"<div><p>Let <span><math><mi>P</mi><mo>:</mo><mo>=</mo><mi>U</mi><mo>(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>)</mo><mo>/</mo><mi>U</mi><mo>(</mo><mi>p</mi><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> be the complex Grassmann manifold and <span><math><mi>F</mi><mo>:</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msup><mi>P</mi><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></math></span> be an arbitrary <span><math><mi>U</mi><mo>(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>)</mo></math></span>-invariant strongly pseudoconvex complex Finsler metric. We prove that <em>F</em> is necessary a Kähler-Berwald metric which is not necessary Hermitian quadratic. We also prove that <em>F</em> is Hermitian quadratic if and only if <em>F</em> is a constant multiple of the canonical <span><math><mi>U</mi><mo>(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>)</mo></math></span>-invariant Kähler metric on <span><math><mi>P</mi></math></span>. In particular on the complex projective space <span><math><msup><mrow><mi>CP</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><mi>U</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mi>U</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>, there exists no <span><math><mi>U</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Fubini-Study metric. These invariant metrics are of particular interesting since they are the most important examples of strongly pseudoconvex complex Finsler metrics on <span><math><mi>P</mi></math></span> which are elliptic metrics in the sense that they enjoy very similar holomorphic sectional curvature and bisectional curvature properties as that of the <span><math><mi>U</mi><mo>(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>)</mo></math></span>-invariant Kähler metrics on <span><math><mi>P</mi></math></span>, nevertheless, these invariant metrics are not necessary Hermitian quadratic, hence provide nontrivial explicit examples for complex Finsler geometry in the compact cases.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"94 ","pages":"Article 102138"},"PeriodicalIF":0.5,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140618046","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 some basic curvature invariants of screen homothetic lightlike hypersurfaces in a GRW spacetime","authors":"Idrees Fayaz Harry ,&nbsp;Mehraj Ahmad Lone ,&nbsp;Alina-Daniela Vîlcu ,&nbsp;Gabriel-Eduard Vîlcu","doi":"10.1016/j.difgeo.2024.102140","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102140","url":null,"abstract":"<div><p>This study is focused on the investigation of lightlike hypersurfaces of a generalized Robertson-Walker (GRW) spacetime. Recently, Poyraz (2022) <span>[51]</span>, <span>[52]</span> established some basic inequalities involving various curvature invariants of screen homothetic lightlike hypersurfaces of GRW spacetimes, like <em>k</em>-scalar curvature and <em>k</em>-Ricci curvature. In this work, we consider other basic curvature invariants, namely the scalar curvature and <em>δ</em>-Casorati curvatures, and derive new inequalities for such hypersurfaces of a GRW spacetime. We also find the conditions for which the equality cases in these inequalities hold and give some applications in Lorentzian geometry.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"94 ","pages":"Article 102140"},"PeriodicalIF":0.5,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140620987","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 Finsler metrics with reversible Douglas curvature","authors":"Guangzu Chen ,&nbsp;Jiayu Liao,&nbsp;Lihong Liu","doi":"10.1016/j.difgeo.2024.102137","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102137","url":null,"abstract":"<div><p>In this paper, we find a new tensor which is responsible for Finsler metrics with reversible geodesics. Using this tensor, we can prove that Finsler metrics are Douglas metrics if and only if they have reversible geodesics and Douglas curvature. Further, we focus on Finsler metrics with reversible Douglas curvature.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"94 ","pages":"Article 102137"},"PeriodicalIF":0.5,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140557439","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":"Gromov–Hausdorff convergence of metric pairs and metric tuples","authors":"Andrés Ahumada Gómez ,&nbsp;Mauricio Che","doi":"10.1016/j.difgeo.2024.102135","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102135","url":null,"abstract":"<div><p>We study the Gromov–Hausdorff convergence of metric pairs and metric tuples and prove the equivalence of different natural definitions of this concept. We also prove embedding, completeness and compactness theorems in this setting. Finally, we get a relative version of Fukaya's theorem about quotient spaces under Gromov–Hausdorff equivariant convergence and a version of Grove–Petersen–Wu's finiteness theorem for stratified spaces.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"94 ","pages":"Article 102135"},"PeriodicalIF":0.5,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0926224524000287/pdfft?md5=90e659088fe8f3dd0f018ed3d1606609&pid=1-s2.0-S0926224524000287-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140342326","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":"Geometry over algebras","authors":"Hugo Cattarucci Botós","doi":"10.1016/j.difgeo.2024.102134","DOIUrl":"https://doi.org/10.1016/j.difgeo.2024.102134","url":null,"abstract":"<div><p>We study geometric structures arising from Hermitian forms on linear spaces over real algebras beyond the division ones. Our focus is on the dual numbers, the split-complex numbers, and the split-quaternions. The corresponding geometric structures are employed to describe the spaces of oriented geodesics in the hyperbolic plane, the Euclidean plane, and the round 2-sphere. We also introduce a simple and natural geometric transition between these spaces. Finally, we present a projective model for the hyperbolic bidisc, that is, the Riemannian product of two hyperbolic discs.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"94 ","pages":"Article 102134"},"PeriodicalIF":0.5,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140320403","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}