Pub Date : 2025-12-01Epub Date: 2025-11-07DOI: 10.1016/j.difgeo.2025.102307
Saajid Chowdhury , Hechen Hu , Matthew Romney , Adam Tsou
We study the properties of surfaces: length metric spaces homeomorphic to a surface having curvature bounded above in the sense of satisfying the condition locally. The main facts about surfaces seem to be largely a part of mathematical folklore, and this paper is intended to rectify the situation. We provide a complete proof that surfaces have bounded (integral) curvature. This fact allows one to apply the established theory of surfaces of bounded curvature to derive further properties of surfaces. We also show that surfaces can be approximated by smooth Riemannian surfaces of Gaussian curvature at most κ. We do this by giving explicit formulas for smoothing the vertices of model polyhedral surfaces.
{"title":"On CAT(κ) surfaces","authors":"Saajid Chowdhury , Hechen Hu , Matthew Romney , Adam Tsou","doi":"10.1016/j.difgeo.2025.102307","DOIUrl":"10.1016/j.difgeo.2025.102307","url":null,"abstract":"<div><div>We study the properties of <span><math><mi>CAT</mi><mo>(</mo><mi>κ</mi><mo>)</mo></math></span> surfaces: length metric spaces homeomorphic to a surface having curvature bounded above in the sense of satisfying the <span><math><mi>CAT</mi><mo>(</mo><mi>κ</mi><mo>)</mo></math></span> condition locally. The main facts about <span><math><mi>CAT</mi><mo>(</mo><mi>κ</mi><mo>)</mo></math></span> surfaces seem to be largely a part of mathematical folklore, and this paper is intended to rectify the situation. We provide a complete proof that <span><math><mi>CAT</mi><mo>(</mo><mi>κ</mi><mo>)</mo></math></span> surfaces have bounded (integral) curvature. This fact allows one to apply the established theory of surfaces of bounded curvature to derive further properties of <span><math><mi>CAT</mi><mo>(</mo><mi>κ</mi><mo>)</mo></math></span> surfaces. We also show that <span><math><mi>CAT</mi><mo>(</mo><mi>κ</mi><mo>)</mo></math></span> surfaces can be approximated by smooth Riemannian surfaces of Gaussian curvature at most <em>κ</em>. We do this by giving explicit formulas for smoothing the vertices of model polyhedral surfaces.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102307"},"PeriodicalIF":0.7,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145465771","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 : 2025-12-01Epub 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":"2025-12-01","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 : 2025-12-01Epub Date: 2025-09-09DOI: 10.1016/j.difgeo.2025.102279
Qing Cui
We show that a closed minimal hypersurface in with constant scalar curvature and zero Gauss curvature is totally geodesic, which gives a support to strong version of Chern conjecture.
{"title":"Minimal hypersurfaces in S5 with constant scalar curvature and zero Gauss curvature","authors":"Qing Cui","doi":"10.1016/j.difgeo.2025.102279","DOIUrl":"10.1016/j.difgeo.2025.102279","url":null,"abstract":"<div><div>We show that a closed minimal hypersurface in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span> with constant scalar curvature and zero Gauss curvature is totally geodesic, which gives a support to strong version of Chern conjecture.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102279"},"PeriodicalIF":0.7,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145020439","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 : 2025-12-01Epub 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":"2025-12-01","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 : 2025-12-01Epub Date: 2025-11-17DOI: 10.1016/j.difgeo.2025.102311
Chi Hin Chan , Magdalena Czubak , Carlos Pinilla Suarez
The Hodge decomposition is well-known for compact manifolds. The result has been extended by Kodaira to include non-compact manifolds and forms. We further extend the Hodge decomposition to the Sobolev space for general k-forms on non-compact manifolds of nonpositive constant sectional curvature. As a result, we also obtain a decomposition on .
{"title":"Hodge decomposition of the Sobolev space H1 on a space form of nonpositive curvature","authors":"Chi Hin Chan , Magdalena Czubak , Carlos Pinilla Suarez","doi":"10.1016/j.difgeo.2025.102311","DOIUrl":"10.1016/j.difgeo.2025.102311","url":null,"abstract":"<div><div>The Hodge decomposition is well-known for compact manifolds. The result has been extended by Kodaira to include non-compact manifolds and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> forms. We further extend the Hodge decomposition to the Sobolev space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> for general <em>k</em>-forms on non-compact manifolds of nonpositive constant sectional curvature. As a result, we also obtain a decomposition on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102311"},"PeriodicalIF":0.7,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145579252","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 : 2025-09-01Epub Date: 2025-06-05DOI: 10.1016/j.difgeo.2025.102263
Tanveer Fatima , Mohammad Shuaib
In this article, we examine conformal hemi-slant submersion from Sasakian manifold onto a Riemannian manifold which generalizes the conformal anti-invariant, conformal semi-invariant and conformal slant submersions and non-trivial examples are provided. We have also covered integrability requirements and address the necessary and sufficient conditions for the totally geodesicness of distributions. Moreover, the sufficient condition for a conformal hemi-slant submersion to be a homothetic map is investigated. The condition for a total manifold of the submersion to be twisted product is also studied, followed by other decomposition theorems.
{"title":"Conformal hemi-slant submersion from Sasakian manifold","authors":"Tanveer Fatima , Mohammad Shuaib","doi":"10.1016/j.difgeo.2025.102263","DOIUrl":"10.1016/j.difgeo.2025.102263","url":null,"abstract":"<div><div>In this article, we examine conformal hemi-slant submersion from Sasakian manifold onto a Riemannian manifold which generalizes the conformal anti-invariant, conformal semi-invariant and conformal slant submersions and non-trivial examples are provided. We have also covered integrability requirements and address the necessary and sufficient conditions for the totally geodesicness of distributions. Moreover, the sufficient condition for a conformal hemi-slant submersion to be a homothetic map is investigated. The condition for a total manifold of the submersion to be twisted product is also studied, followed by other decomposition theorems.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102263"},"PeriodicalIF":0.6,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144222617","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 : 2025-09-01Epub Date: 2025-07-15DOI: 10.1016/j.difgeo.2025.102270
D. Bayril , J.M. Selig
In this work the kinematic geometry of line-symmetric rigid-body motions is revisited. These motions are produced by reflecting a rigid body in the successive generator lines of a ruled surface. Classical results are re-derived using methods from Lie algebra and new results are found. In particular, results for some of the acceleration properties of these motions are found using the Sannia frame of the ruled surfaces. The ruled surfaces considered are given by the tangent, normal or binormal lines to smooth curves as well as Catalan surfaces and right conoids.
{"title":"The geometry of line-symmetric rigid-body motions","authors":"D. Bayril , J.M. Selig","doi":"10.1016/j.difgeo.2025.102270","DOIUrl":"10.1016/j.difgeo.2025.102270","url":null,"abstract":"<div><div>In this work the kinematic geometry of line-symmetric rigid-body motions is revisited. These motions are produced by reflecting a rigid body in the successive generator lines of a ruled surface. Classical results are re-derived using methods from Lie algebra and new results are found. In particular, results for some of the acceleration properties of these motions are found using the Sannia frame of the ruled surfaces. The ruled surfaces considered are given by the tangent, normal or binormal lines to smooth curves as well as Catalan surfaces and right conoids.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102270"},"PeriodicalIF":0.6,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144631355","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 : 2025-09-01Epub Date: 2025-08-25DOI: 10.1016/j.difgeo.2025.102277
Noriaki Ikeda
We propose a Hamiltonian Lie algebroid and a momentum section over a Dirac structure as a generalization of a Hamiltonian Lie algebroid over a pre-symplectic manifold and one over a Poisson manifold. A Hamiltonian Lie algebroid and a momentum section generalize a Hamiltonian G-space and a momentum map over a symplectic manifold. We prove some properties of this new Hamiltonian Lie algebroid and construct a mechanics based on this structure as an application. These new mechanics are called the gauged Poisson sigma model and the gauged Dirac sigma model.
{"title":"Hamilton Lie algebroids over Dirac structures and sigma models","authors":"Noriaki Ikeda","doi":"10.1016/j.difgeo.2025.102277","DOIUrl":"10.1016/j.difgeo.2025.102277","url":null,"abstract":"<div><div>We propose a Hamiltonian Lie algebroid and a momentum section over a Dirac structure as a generalization of a Hamiltonian Lie algebroid over a pre-symplectic manifold and one over a Poisson manifold. A Hamiltonian Lie algebroid and a momentum section generalize a Hamiltonian <em>G</em>-space and a momentum map over a symplectic manifold. We prove some properties of this new Hamiltonian Lie algebroid and construct a mechanics based on this structure as an application. These new mechanics are called the gauged Poisson sigma model and the gauged Dirac sigma model.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102277"},"PeriodicalIF":0.7,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144893640","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 : 2025-09-01Epub Date: 2025-07-09DOI: 10.1016/j.difgeo.2025.102269
Andrew James Bruce , Janusz Grabowski
We introduce and examine the notion of principal -bundles, i.e., principal bundles in the category of -manifolds. The latter are higher graded extensions of supermanifolds in which a -grading replaces -grading. These extensions have opened up new areas of research of great interest in both physics and mathematics. In principle, the geometry of -manifolds is essentially different than that of supermanifolds, as for n > 1 we have formal variables of even parity, so local smooth functions are power series in formal variables. On the other hand, a full version of differential calculus is still valid. We show in this paper that the fundamental properties of classical principal bundles can be generalised to the setting of this ‘higher graded’ geometry, with properly defined frame bundles of -vector bundles as canonical examples. Additionally, we propose a new approach to the concept of a vector bundle in this setting. However, formulating these ideas and proving these results relies on many technical upshots established in earlier papers. A comprehensive introduction to -manifolds is therefore included together with basic examples.
{"title":"Principal bundles in the category of Z2n-manifolds","authors":"Andrew James Bruce , Janusz Grabowski","doi":"10.1016/j.difgeo.2025.102269","DOIUrl":"10.1016/j.difgeo.2025.102269","url":null,"abstract":"<div><div>We introduce and examine the notion of principal <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-bundles, i.e., principal bundles in the category of <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-manifolds. The latter are higher graded extensions of supermanifolds in which a <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-grading replaces <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-grading. These extensions have opened up new areas of research of great interest in both physics and mathematics. In principle, the geometry of <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-manifolds is essentially different than that of supermanifolds, as for n > 1 we have formal variables of even parity, so local smooth functions are power series in formal variables. On the other hand, a full version of differential calculus is still valid. We show in this paper that the fundamental properties of classical principal bundles can be generalised to the setting of this ‘higher graded’ geometry, with properly defined frame bundles of <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-vector bundles as canonical examples. Additionally, we propose a new approach to the concept of a vector bundle in this setting. However, formulating these ideas and proving these results relies on many technical upshots established in earlier papers. A comprehensive introduction to <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-manifolds is therefore included together with basic examples.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102269"},"PeriodicalIF":0.6,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579541","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 : 2025-09-01Epub Date: 2025-08-20DOI: 10.1016/j.difgeo.2025.102275
Kurando Baba , Kazuto Shintani
F-Yang-Mills connections are critical points of F-Yang Mills functional on the space of connections of a principal fiber bundle, which are generalizations of Yang-Mills connections, p-Yang-Mills connections and exponential Yang-Mills connections and so on. Here, F is a strictly increasing -function. In this paper, we extend Simons theorem for the instability of Yang-Mills connections to F-Yang-Mills connections. We derive a sufficient condition that any non-flat, F-Yang-Mills connection over submanifolds in a Euclidean space is unstable. In the standard sphere case, this condition is expressed by an inequality involving its dimension and the degree of the differential of the function F. Our main results are proved by extending Kobayashi-Ohnita-Takeuchi's calculation to F-Yang-Mills connections.
{"title":"A Simons type condition for instability of F-Yang-Mills connections","authors":"Kurando Baba , Kazuto Shintani","doi":"10.1016/j.difgeo.2025.102275","DOIUrl":"10.1016/j.difgeo.2025.102275","url":null,"abstract":"<div><div><em>F</em>-Yang-Mills connections are critical points of <em>F</em>-Yang Mills functional on the space of connections of a principal fiber bundle, which are generalizations of Yang-Mills connections, <em>p</em>-Yang-Mills connections and exponential Yang-Mills connections and so on. Here, <em>F</em> is a strictly increasing <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-function. In this paper, we extend Simons theorem for the instability of Yang-Mills connections to <em>F</em>-Yang-Mills connections. We derive a sufficient condition that any non-flat, <em>F</em>-Yang-Mills connection over submanifolds in a Euclidean space is unstable. In the standard sphere case, this condition is expressed by an inequality involving its dimension and the degree of the differential of the function <em>F</em>. Our main results are proved by extending Kobayashi-Ohnita-Takeuchi's calculation to <em>F</em>-Yang-Mills connections.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102275"},"PeriodicalIF":0.7,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144867305","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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