Pub Date : 2025-10-01DOI: 10.1016/j.difgeo.2025.102300
Ioana Ciuclea, Cornelia Vizman
We study a class of coadjoint orbits of the area preserving diffeomorphism group of the plane consisting of vortex loops, namely closed curves in the plane decorated with one-forms (vorticity densities) allowed to have zeros.
{"title":"Manifolds of vortex loops as coadjoint orbits","authors":"Ioana Ciuclea, Cornelia Vizman","doi":"10.1016/j.difgeo.2025.102300","DOIUrl":"10.1016/j.difgeo.2025.102300","url":null,"abstract":"<div><div>We study a class of coadjoint orbits of the area preserving diffeomorphism group of the plane consisting of vortex loops, namely closed curves in the plane decorated with one-forms (vorticity densities) allowed to have zeros.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102300"},"PeriodicalIF":0.7,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145221240","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}
Pub Date : 2025-09-26DOI: 10.1016/j.difgeo.2025.102299
Wei Wang
The k-Cauchy-Fueter complex in quaternionic analysis is the counterpart of the Dolbeault complex in complex analysis. In this paper, we find the explicit transformation formula of these complexes under , which acts on as quaternionic fractional linear transformations. These transformation formulae have several interesting applications to k-regular functions, the quaternionic counterpart of holomorphic functions, and geometry of domains. They allow us to construct the k-Cauchy-Fueter complex over locally quaternionic projective flat manifolds explicitly and introduce various notions of pluripotential theory on this kind of manifolds. We also construct a quaternionic projectively invariant operator from the quaternionic Monge-Ampère operator, which can be used to find projectively invariant defining density of a domain, generalizing Fefferman's construction in complex analysis.
{"title":"Quaternionic projective invariance of the k-Cauchy-Fueter complex and applications I","authors":"Wei Wang","doi":"10.1016/j.difgeo.2025.102299","DOIUrl":"10.1016/j.difgeo.2025.102299","url":null,"abstract":"<div><div>The <em>k</em>-Cauchy-Fueter complex in quaternionic analysis is the counterpart of the Dolbeault complex in complex analysis. In this paper, we find the explicit transformation formula of these complexes under <span><math><mrow><mi>SL</mi></mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>H</mi><mo>)</mo></math></span>, which acts on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> as quaternionic fractional linear transformations. These transformation formulae have several interesting applications to <em>k</em>-regular functions, the quaternionic counterpart of holomorphic functions, and geometry of domains. They allow us to construct the <em>k</em>-Cauchy-Fueter complex over locally quaternionic projective flat manifolds explicitly and introduce various notions of pluripotential theory on this kind of manifolds. We also construct a quaternionic projectively invariant operator from the quaternionic Monge-Ampère operator, which can be used to find projectively invariant defining density of a domain, generalizing Fefferman's construction in complex analysis.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102299"},"PeriodicalIF":0.7,"publicationDate":"2025-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145158564","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}
Pub Date : 2025-09-25DOI: 10.1016/j.difgeo.2025.102298
Paula Carretero , Ildefonso Castro
We solve the problem of prescribing different types of curvatures (principal, mean or Gaussian) on rotational surfaces in terms of arbitrary continuous functions depending on the distance from the surface to the axis of revolution. In this line, we get the complete explicit classification of the rotational surfaces with mean or Gauss curvature inversely proportional to the distance from the surface to the axis of revolution. We also provide new uniqueness results on some well known surfaces, such as the catenoid or the torus of revolution, and others less well known but equally interesting for their physical applications, such as the Mylar balloon or the Flamm's paraboloid.
{"title":"Rotational surfaces with prescribed curvatures","authors":"Paula Carretero , Ildefonso Castro","doi":"10.1016/j.difgeo.2025.102298","DOIUrl":"10.1016/j.difgeo.2025.102298","url":null,"abstract":"<div><div>We solve the problem of prescribing different types of curvatures (principal, mean or Gaussian) on rotational surfaces in terms of arbitrary continuous functions depending on the distance from the surface to the axis of revolution. In this line, we get the complete explicit classification of the rotational surfaces with mean or Gauss curvature inversely proportional to the distance from the surface to the axis of revolution. We also provide new uniqueness results on some well known surfaces, such as the catenoid or the torus of revolution, and others less well known but equally interesting for their physical applications, such as the Mylar balloon or the Flamm's paraboloid.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102298"},"PeriodicalIF":0.7,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145158565","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}
Pub Date : 2025-09-22DOI: 10.1016/j.difgeo.2025.102295
Valeria Gutiérrez
We consider the homogeneous space , where is an irreducible symmetric space and ΔK denotes diagonal embedding. Recently, Lauret and Will provided a complete classification of -invariant Einstein metrics on M. They obtained that there is always at least one non-diagonal Einstein metric on M, and in some cases, diagonal Einstein metrics also exist. We give a formula for the scalar curvature of a subset of -invariant metrics and study the stability of non-diagonal Einstein metrics on M with respect to the Hilbert action, obtaining that these metrics are unstable with different coindices for all homogeneous spaces M.
{"title":"Stability of non-diagonal Einstein metrics on homogeneous spaces H × H/ΔK","authors":"Valeria Gutiérrez","doi":"10.1016/j.difgeo.2025.102295","DOIUrl":"10.1016/j.difgeo.2025.102295","url":null,"abstract":"<div><div>We consider the homogeneous space <span><math><mi>M</mi><mo>=</mo><mi>H</mi><mo>×</mo><mi>H</mi><mo>/</mo><mi>Δ</mi><mi>K</mi></math></span>, where <span><math><mi>H</mi><mo>/</mo><mi>K</mi></math></span> is an irreducible symmetric space and Δ<em>K</em> denotes diagonal embedding. Recently, Lauret and Will provided a complete classification of <span><math><mi>H</mi><mo>×</mo><mi>H</mi></math></span>-invariant Einstein metrics on M. They obtained that there is always at least one non-diagonal Einstein metric on <em>M</em>, and in some cases, diagonal Einstein metrics also exist. We give a formula for the scalar curvature of a subset of <span><math><mi>H</mi><mo>×</mo><mi>H</mi></math></span>-invariant metrics and study the stability of non-diagonal Einstein metrics on <em>M</em> with respect to the Hilbert action, obtaining that these metrics are unstable with different coindices for all homogeneous spaces <em>M</em>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102295"},"PeriodicalIF":0.7,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145109231","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}
Pub Date : 2025-09-19DOI: 10.1016/j.difgeo.2025.102297
Nasrin Sadeghzadeh, Meshkat Yavari
The development of projective invariant Weyl metrics in this paper offers a fresh perspective, as we establish the characteristics of both weakly-Weyl and generalized weakly-Weyl Finsler metrics. We thoroughly examine the connections between these metrics and various projective invariants, highlighting their significance in the context of generalized Sakaguchi's Theorem, which states that every Finsler metric of scalar flag curvature is a GDW-metric. Additionally, we introduce several illustrative examples pertaining to this new class of projective invariant Finsler metrics. Specifically, we explore the category of weakly-Weyl spherically symmetric Finsler metrics in . Importantly, we demonstrate that the two classes weakly-Weyl and W-quadratic spherically symmetric Finsler metrics in are equivalent.
{"title":"Generalized weakly-Weyl Finsler metrics: A generalized approach to Sakaguchi's theorem","authors":"Nasrin Sadeghzadeh, Meshkat Yavari","doi":"10.1016/j.difgeo.2025.102297","DOIUrl":"10.1016/j.difgeo.2025.102297","url":null,"abstract":"<div><div>The development of projective invariant Weyl metrics in this paper offers a fresh perspective, as we establish the characteristics of both weakly-Weyl and generalized weakly-Weyl Finsler metrics. We thoroughly examine the connections between these metrics and various projective invariants, highlighting their significance in the context of generalized Sakaguchi's Theorem, which states that every Finsler metric of scalar flag curvature is a GDW-metric. Additionally, we introduce several illustrative examples pertaining to this new class of projective invariant Finsler metrics. Specifically, we explore the category of weakly-Weyl spherically symmetric Finsler metrics in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Importantly, we demonstrate that the two classes weakly-Weyl and <em>W</em>-quadratic spherically symmetric Finsler metrics in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> are equivalent.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102297"},"PeriodicalIF":0.7,"publicationDate":"2025-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145106321","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}
Pub Date : 2025-09-19DOI: 10.1016/j.difgeo.2025.102293
Andreas Schüßler
We give a proof in the context of smooth differential geometry of Polishchuk's theorem from 1997, in which he established under which conditions, given a Poisson scheme M and a Poisson subscheme N, the Poisson structure lifts to the blowup of M along N.
{"title":"On the real projective blowup of Poisson structures","authors":"Andreas Schüßler","doi":"10.1016/j.difgeo.2025.102293","DOIUrl":"10.1016/j.difgeo.2025.102293","url":null,"abstract":"<div><div>We give a proof in the context of smooth differential geometry of Polishchuk's theorem from 1997, in which he established under which conditions, given a Poisson scheme <em>M</em> and a Poisson subscheme <em>N</em>, the Poisson structure lifts to the blowup of <em>M</em> along <em>N</em>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102293"},"PeriodicalIF":0.7,"publicationDate":"2025-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145106320","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}
Pub Date : 2025-09-18DOI: 10.1016/j.difgeo.2025.102296
Andreas Schüßler, Marco Zambon
Lie algebroids, singular foliations, and Dirac structures are closely related objects. We examine the relation between their pullbacks under maps satisfying a constant rank or transversality assumption. A special case is given by blowdown maps. In that case, we also establish the relation between the blowup of a Lie algebroid and its singular foliation.
{"title":"A note on pullbacks and blowups of Lie algebroids, singular foliations, and Dirac structures","authors":"Andreas Schüßler, Marco Zambon","doi":"10.1016/j.difgeo.2025.102296","DOIUrl":"10.1016/j.difgeo.2025.102296","url":null,"abstract":"<div><div>Lie algebroids, singular foliations, and Dirac structures are closely related objects. We examine the relation between their pullbacks under maps satisfying a constant rank or transversality assumption. A special case is given by blowdown maps. In that case, we also establish the relation between the blowup of a Lie algebroid and its singular foliation.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102296"},"PeriodicalIF":0.7,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145106184","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}
Pub Date : 2025-09-16DOI: 10.1016/j.difgeo.2025.102294
Ming Wu , Ju Tan , Na Xu
In this article, we obtain that the compact simple Lie group admits at least two new non-naturally reductive -invariant Einstein metrics, and the compact simple Lie group admits at least two new non-naturally reductive -invariant Einstein metrics. Furthermore, we explore the isometry problem for these Einstein metrics. Finally, we prove that there are at least two families of non-naturally reductive invariant Einstein-Randers metrics on and respectively.
{"title":"Non-naturally reductive Einstein metrics on SO(n)","authors":"Ming Wu , Ju Tan , Na Xu","doi":"10.1016/j.difgeo.2025.102294","DOIUrl":"10.1016/j.difgeo.2025.102294","url":null,"abstract":"<div><div>In this article, we obtain that the compact simple Lie group <span><math><mi>S</mi><mi>O</mi><mo>(</mo><mn>4</mn><mi>k</mi><mo>+</mo><mn>3</mn><mo>)</mo><mo>(</mo><mi>k</mi><mo>≥</mo><mn>8</mn><mo>)</mo></math></span> admits at least two new non-naturally reductive <span><math><mi>A</mi><mi>d</mi><mo>(</mo><mi>S</mi><mi>O</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>×</mo><mi>S</mi><mi>O</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>×</mo><mi>S</mi><mi>O</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>×</mo><mi>S</mi><mi>O</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>)</mo></math></span>-invariant Einstein metrics, and the compact simple Lie group <span><math><mi>S</mi><mi>O</mi><mo>(</mo><mn>4</mn><mi>k</mi><mo>+</mo><mn>6</mn><mo>)</mo><mo>(</mo><mi>k</mi><mo>≥</mo><mn>11</mn><mo>)</mo></math></span> admits at least two new non-naturally reductive <span><math><mi>A</mi><mi>d</mi><mo>(</mo><mi>S</mi><mi>O</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>×</mo><mi>S</mi><mi>O</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>×</mo><mi>S</mi><mi>O</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>×</mo><mi>S</mi><mi>O</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>)</mo></math></span>-invariant Einstein metrics. Furthermore, we explore the isometry problem for these Einstein metrics. Finally, we prove that there are at least two families of non-naturally reductive invariant Einstein-Randers metrics on <span><math><mi>S</mi><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>(</mo><mi>n</mi><mo>≥</mo><mn>35</mn><mo>)</mo></math></span> and <span><math><mi>S</mi><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>(</mo><mi>n</mi><mo>≥</mo><mn>50</mn><mo>)</mo></math></span> respectively.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102294"},"PeriodicalIF":0.7,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145106185","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}
Pub Date : 2025-09-16DOI: 10.1016/j.difgeo.2025.102292
Juan Sebastián Herrera-Carmona , Cristian Ortiz , James Waldron
We introduce and study module structures on both the dgla of multiplicative vector fields and the graded algebra of functions on Lie groupoids. We show that there is an associated structure of a graded Lie-Rinehart algebra on the vector fields of a differentiable stack over its smooth functions that is Morita invariant in an appropriate sense. Furthermore, we show that associated Van-Est type maps are compatible with those module structures. We also present several examples.
{"title":"Vector fields and derivations on differentiable stacks","authors":"Juan Sebastián Herrera-Carmona , Cristian Ortiz , James Waldron","doi":"10.1016/j.difgeo.2025.102292","DOIUrl":"10.1016/j.difgeo.2025.102292","url":null,"abstract":"<div><div>We introduce and study module structures on both the dgla of multiplicative vector fields and the graded algebra of functions on Lie groupoids. We show that there is an associated structure of a graded Lie-Rinehart algebra on the vector fields of a differentiable stack over its smooth functions that is Morita invariant in an appropriate sense. Furthermore, we show that associated Van-Est type maps are compatible with those module structures. We also present several examples.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102292"},"PeriodicalIF":0.7,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145106183","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}
Pub Date : 2025-09-12DOI: 10.1016/j.difgeo.2025.102291
Diego Artacho, Marie-Amélie Lawn
Spinorial methods have proven to be a powerful tool to study geometric properties of spin manifolds. Our aim is to continue the spinorial study of manifolds that are not necessarily spin. We introduce and study the notion of G-invariance of spinr structures on a manifold M equipped with an action of a Lie group G. For the case when M is a homogeneous G-space, we prove a classification result of these invariant structures in terms of the isotropy representation. As an example, we study the invariant spinr structures for all the homogeneous realisations of the spheres.
{"title":"Generalised spinr structures on homogeneous spaces","authors":"Diego Artacho, Marie-Amélie Lawn","doi":"10.1016/j.difgeo.2025.102291","DOIUrl":"10.1016/j.difgeo.2025.102291","url":null,"abstract":"<div><div>Spinorial methods have proven to be a powerful tool to study geometric properties of spin manifolds. Our aim is to continue the spinorial study of manifolds that are not necessarily spin. We introduce and study the notion of <em>G</em>-invariance of spin<sup><em>r</em></sup> structures on a manifold <em>M</em> equipped with an action of a Lie group <em>G</em>. For the case when <em>M</em> is a homogeneous <em>G</em>-space, we prove a classification result of these invariant structures in terms of the isotropy representation. As an example, we study the invariant spin<sup><em>r</em></sup> structures for all the homogeneous realisations of the spheres.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102291"},"PeriodicalIF":0.7,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049118","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}
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