Pub 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":"2025-07-15","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 : 2025-07-15DOI: 10.1016/j.difgeo.2025.102271
Shinji Ohno , Yuuki Sasaki
Cohomogeneity one actions on irreducible Riemannian symmetric spaces of compact type are classified into three cases: Hermann actions, actions induced by the linear isotropy representation of a Riemannian symmetric space of rank 2, and exceptional actions. In this paper, we consider exceptional actions related to the exceptional compact Lie group and investigate some properties of their orbits as Riemannian submanifolds. In particular, we examine the principal curvatures of principal orbits and classify principal orbits that are minimal, austere, weakly reflective, and proper biharmonic.
{"title":"On cohomogeneity one hyperpolar actions related to G2","authors":"Shinji Ohno , Yuuki Sasaki","doi":"10.1016/j.difgeo.2025.102271","DOIUrl":"10.1016/j.difgeo.2025.102271","url":null,"abstract":"<div><div>Cohomogeneity one actions on irreducible Riemannian symmetric spaces of compact type are classified into three cases: Hermann actions, actions induced by the linear isotropy representation of a Riemannian symmetric space of rank 2, and exceptional actions. In this paper, we consider exceptional actions related to the exceptional compact Lie group <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and investigate some properties of their orbits as Riemannian submanifolds. In particular, we examine the principal curvatures of principal orbits and classify principal orbits that are minimal, austere, weakly reflective, and proper biharmonic.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102271"},"PeriodicalIF":0.6,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144631296","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 : 2025-07-09DOI: 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":"2025-07-09","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 : 2025-07-09DOI: 10.1016/j.difgeo.2025.102265
Alberto S. Cattaneo, Shuhan Jiang
We derive equivariant localization formulas of Atiyah–Bott and cohomological field theory types in the Batalin-Vilkovisky formalism and discuss their applications in Poisson geometry and quantum field theory.
{"title":"Equivariant localization in Batalin-Vilkovisky formalism","authors":"Alberto S. Cattaneo, Shuhan Jiang","doi":"10.1016/j.difgeo.2025.102265","DOIUrl":"10.1016/j.difgeo.2025.102265","url":null,"abstract":"<div><div>We derive equivariant localization formulas of Atiyah–Bott and cohomological field theory types in the Batalin-Vilkovisky formalism and discuss their applications in Poisson geometry and quantum field theory.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102265"},"PeriodicalIF":0.6,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579540","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 : 2025-07-02DOI: 10.1016/j.difgeo.2025.102268
Chang-Jian Zhao
In the paper, our main aim is to generalize the dual mixed harmonic quermassintegrals to Orlicz space. Under the framework of Orlicz dual Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating Orlicz first order variation of the dual mixed harmonic quermassintegrals, and call it the Orlicz dual mixed harmonic quermassintegrals. The fundamental notions and conclusions of the dual mixed harmonic quermassintegrals and the Minkowski and Brunn-Minkowski inequalities for the dual harmonic quermassintegrals are extended to an Orlicz setting, and the related concepts and inequalities of Orlicz dual mixed volumes are also included in our conclusions.
{"title":"Orlicz harmonic version of dual mixed volumes","authors":"Chang-Jian Zhao","doi":"10.1016/j.difgeo.2025.102268","DOIUrl":"10.1016/j.difgeo.2025.102268","url":null,"abstract":"<div><div>In the paper, our main aim is to generalize the dual mixed harmonic quermassintegrals to Orlicz space. Under the framework of Orlicz dual Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating Orlicz first order variation of the dual mixed harmonic quermassintegrals, and call it the Orlicz dual mixed harmonic quermassintegrals. The fundamental notions and conclusions of the dual mixed harmonic quermassintegrals and the Minkowski and Brunn-Minkowski inequalities for the dual harmonic quermassintegrals are extended to an Orlicz setting, and the related concepts and inequalities of Orlicz dual mixed volumes are also included in our conclusions.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102268"},"PeriodicalIF":0.6,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144534227","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 : 2025-06-25DOI: 10.1016/j.difgeo.2025.102267
Nan Li, Zhenghan Shen
In this paper, we give some vanishing theorems for p-harmonic l-forms on complete non-compact Riemannian manifolds. Firstly, we prove a vanishing theorem on Riemannian manifolds with nonnegative scalar curvature and a more general upper bound of pointwise curvature condition. Secondly, by using the similar trick, we obtain some vanishing theorems on complete immersed submanifold of Euclidean space.
{"title":"Some vanishing theorems for p-harmonic l-forms on complete Riemannian manifolds","authors":"Nan Li, Zhenghan Shen","doi":"10.1016/j.difgeo.2025.102267","DOIUrl":"10.1016/j.difgeo.2025.102267","url":null,"abstract":"<div><div>In this paper, we give some vanishing theorems for <em>p</em>-harmonic <em>l</em>-forms on complete non-compact Riemannian manifolds. Firstly, we prove a vanishing theorem on Riemannian manifolds with nonnegative scalar curvature and a more general upper bound of pointwise curvature condition. Secondly, by using the similar trick, we obtain some vanishing theorems on complete immersed submanifold of Euclidean space.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102267"},"PeriodicalIF":0.6,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144471724","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 : 2025-06-25DOI: 10.1016/j.difgeo.2025.102266
Alaa Boukholkhal
For any compact Riemannian manifold and any Lorentzian manifold , we prove that any spacelike embedding that is long () can be -approximated by a isometric embedding .
{"title":"C1-isometric embeddings of Riemannian spaces in Lorentzian spaces","authors":"Alaa Boukholkhal","doi":"10.1016/j.difgeo.2025.102266","DOIUrl":"10.1016/j.difgeo.2025.102266","url":null,"abstract":"<div><div>For any compact Riemannian manifold <span><math><mo>(</mo><mi>V</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> and any Lorentzian manifold <span><math><mo>(</mo><mi>W</mi><mo>,</mo><mi>h</mi><mo>)</mo></math></span>, we prove that any spacelike embedding <span><math><mi>f</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>W</mi></math></span> that is long (<span><math><mi>g</mi><mo>≤</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>h</mi></math></span>) can be <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span>-approximated by a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> isometric embedding <span><math><mi>F</mi><mo>:</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>g</mi><mo>)</mo><mo>→</mo><mo>(</mo><mi>W</mi><mo>,</mo><mi>h</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102266"},"PeriodicalIF":0.6,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144471655","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 : 2025-06-09DOI: 10.1016/j.difgeo.2025.102264
Zisu Zhao
Employing a new Laplacian comparison theorem, we have derived a Souplet-Zhang type gradient estimate for a specific nonlinear parabolic equation (Finslerian logarithmic Schrödinger equation) on a non-compact forward complete Finsler manifold with some curvatures bounded from below. All the coefficients in our equations vary with time on the manifold. As applications, we obtain a local Harnack inequality and a Liouville-type theorem.
{"title":"A Hamilton-Souplet-Zhang type gradient estimate for a class of parabolic equations on Finsler manifolds","authors":"Zisu Zhao","doi":"10.1016/j.difgeo.2025.102264","DOIUrl":"10.1016/j.difgeo.2025.102264","url":null,"abstract":"<div><div>Employing a new Laplacian comparison theorem, we have derived a Souplet-Zhang type gradient estimate for a specific nonlinear parabolic equation (Finslerian logarithmic Schrödinger equation) on a non-compact forward complete Finsler manifold with some curvatures bounded from below. All the coefficients in our equations vary with time on the manifold. As applications, we obtain a local Harnack inequality and a Liouville-type theorem.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102264"},"PeriodicalIF":0.6,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144240986","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 : 2025-06-05DOI: 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":"2025-06-05","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 : 2025-06-02DOI: 10.1016/j.difgeo.2025.102262
Yasha Savelyev
Let be as in the classical Gromov non-squeezing theorem, and let . We first conjecture that the Gromov non-squeezing phenomenon persists for deformations of the symplectic form on the range (w.r.t. the standard metric) ϵ-nearby to the standard symplectic form. We prove this in some special cases, in particular when the dimension is four and when . Given such a perturbation, we can no longer compactify the range and hence the classical Gromov argument breaks down. Our main method consists of a certain trap idea for holomorphic curves, analogous to traps in dynamical systems.
{"title":"A remark on deformation of Gromov non-squeezing","authors":"Yasha Savelyev","doi":"10.1016/j.difgeo.2025.102262","DOIUrl":"10.1016/j.difgeo.2025.102262","url":null,"abstract":"<div><div>Let <span><math><mi>R</mi><mo>,</mo><mi>r</mi></math></span> be as in the classical Gromov non-squeezing theorem, and let <span><math><mi>ϵ</mi><mo>=</mo><mo>(</mo><mi>π</mi><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>π</mi><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>/</mo><mi>π</mi><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. We first conjecture that the Gromov non-squeezing phenomenon persists for deformations of the symplectic form on the range <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> (w.r.t. the standard metric) <em>ϵ</em>-nearby to the standard symplectic form. We prove this in some special cases, in particular when the dimension is four and when <span><math><mi>R</mi><mo><</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mi>r</mi></math></span>. Given such a perturbation, we can no longer compactify the range and hence the classical Gromov argument breaks down. Our main method consists of a certain trap idea for holomorphic curves, analogous to traps in dynamical systems.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102262"},"PeriodicalIF":0.6,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144190117","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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