Pub Date : 2026-01-28DOI: 10.1016/j.difgeo.2026.102345
Mirjana Milijević , Sara Miri
We define and study locally Levi-flat CR statistical submanifolds of maximal CR dimension within holomorphic statistical manifolds of constant holomorphic sectional curvature. Our definition generalizes the corresponding notion in Kähler geometry. Moreover, we establish a relationship between Levi-flatness and holomorphic sectional curvature. In particular, we prove that if a locally Levi-flat CR statistical submanifold admits a certain geometric configuration, namely, when a vector field derived from the complex structure is an eigenvector of both shape operators, then the ambient curvature must be strictly negative.
{"title":"Locally Levi-flat statistical submanifolds","authors":"Mirjana Milijević , Sara Miri","doi":"10.1016/j.difgeo.2026.102345","DOIUrl":"10.1016/j.difgeo.2026.102345","url":null,"abstract":"<div><div>We define and study locally Levi-flat CR statistical submanifolds of maximal CR dimension within holomorphic statistical manifolds of constant holomorphic sectional curvature. Our definition generalizes the corresponding notion in Kähler geometry. Moreover, we establish a relationship between Levi-flatness and holomorphic sectional curvature. In particular, we prove that if a locally Levi-flat CR statistical submanifold admits a certain geometric configuration, namely, when a vector field derived from the complex structure is an eigenvector of both shape operators, then the ambient curvature must be strictly negative.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102345"},"PeriodicalIF":0.7,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079471","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 : 2026-01-16DOI: 10.1016/j.difgeo.2026.102334
Laura Leski , Leonid Ryvkin
We give an intrinsic characterization of multisymplectic manifolds that have the linear type of density-valued symplectic forms in each tangent space, prove Darboux-type theorems for these forms, and investigate their symmetries.
{"title":"Density-valued symplectic forms from a multisymplectic viewpoint","authors":"Laura Leski , Leonid Ryvkin","doi":"10.1016/j.difgeo.2026.102334","DOIUrl":"10.1016/j.difgeo.2026.102334","url":null,"abstract":"<div><div>We give an intrinsic characterization of multisymplectic manifolds that have the linear type of density-valued symplectic forms in each tangent space, prove Darboux-type theorems for these forms, and investigate their symmetries.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102334"},"PeriodicalIF":0.7,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981243","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 : 2026-01-16DOI: 10.1016/j.difgeo.2026.102335
Marius Crainic , João Nuno Mestre
In this note we revisit the notions of transverse density bundle and of modular classes of Lie algebroids and Lie groupoids; in particular, we point out that one should use the transverse density bundle instead of , which is the representation that is commonly used when talking about modular classes. One of the reasons for this is that, as we will see, is not really an object associated with the stack presented by a Lie groupoid (in general, it is not a representation of the groupoid!).
We provide a simple construction of the representation of a Lie groupoid on its transverse volume, orientation, and density bundles in terms of (good) functors on vector spaces. We also extend the modular class by a Stiefel-Whitney class that controls the transverse orientability of a Lie groupoid
{"title":"The transverse density bundle and modular classes of Lie groupoids","authors":"Marius Crainic , João Nuno Mestre","doi":"10.1016/j.difgeo.2026.102335","DOIUrl":"10.1016/j.difgeo.2026.102335","url":null,"abstract":"<div><div>In this note we revisit the notions of transverse density bundle and of modular classes of Lie algebroids and Lie groupoids; in particular, we point out that one should use the transverse density bundle <span><math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>A</mi></mrow><mrow><mtext>tr</mtext></mrow></msubsup></math></span> instead of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span>, which is the representation that is commonly used when talking about modular classes. One of the reasons for this is that, as we will see, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span> is not really an object associated with the stack presented by a Lie groupoid (in general, it is not a representation of the groupoid!).</div><div>We provide a simple construction of the representation of a Lie groupoid on its transverse volume, orientation, and density bundles in terms of (good) functors on vector spaces. We also extend the modular class by a Stiefel-Whitney class that controls the transverse orientability of a Lie groupoid</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102335"},"PeriodicalIF":0.7,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981242","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 : 2026-01-14DOI: 10.1016/j.difgeo.2025.102329
Guanghan Li, Yifan Yang
In this paper, the Minkowski formula and the Heintze-Karcher inequality are obtained for hypersurfaces with capillary boundary in a cone. Then we study a type of inverse mean curvature flow in a cone, as well as its long-time existence and convergence. As a result we derive the capillary isoperimetric inequality for mean convex star-shaped hypersurfaces with capillary boundary in a cone.
{"title":"A flow method to isoperimetric inequality for mean convex star-shaped capillary hypersurfaces in a cone","authors":"Guanghan Li, Yifan Yang","doi":"10.1016/j.difgeo.2025.102329","DOIUrl":"10.1016/j.difgeo.2025.102329","url":null,"abstract":"<div><div>In this paper, the Minkowski formula and the Heintze-Karcher inequality are obtained for hypersurfaces with capillary boundary in a cone. Then we study a type of inverse mean curvature flow in a cone, as well as its long-time existence and convergence. As a result we derive the capillary isoperimetric inequality for mean convex star-shaped hypersurfaces with capillary boundary in a cone.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102329"},"PeriodicalIF":0.7,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145957878","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 : 2026-01-14DOI: 10.1016/j.difgeo.2026.102332
Farida Ghazawneh
Kennard, Khalili Samani, and Searle showed that for a -torus, , acting on a closed, positively curved Riemannian n-manifold, , with a non-empty fixed point set for n large enough and r approximately half the dimension of M, then is homotopy equivalent to , , , or a lens space. In this paper, we lower r to approximately and show that we still obtain the same result.
{"title":"Z2-torus actions on positively curved manifolds","authors":"Farida Ghazawneh","doi":"10.1016/j.difgeo.2026.102332","DOIUrl":"10.1016/j.difgeo.2026.102332","url":null,"abstract":"<div><div>Kennard, Khalili Samani, and Searle showed that for a <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-torus, <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>, acting on a closed, positively curved Riemannian <em>n</em>-manifold, <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, with a non-empty fixed point set for <em>n</em> large enough and <em>r</em> approximately half the dimension of <em>M</em>, then <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is homotopy equivalent to <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mi>R</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span>, or a lens space. In this paper, we lower <em>r</em> to approximately <span><math><mn>2</mn><mi>n</mi><mo>/</mo><mn>5</mn></math></span> and show that we still obtain the same result.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102332"},"PeriodicalIF":0.7,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981244","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 : 2026-01-14DOI: 10.1016/j.difgeo.2026.102333
Marco Castrillón López , Pedro M. Gadea , Boris P. Komrakov , M. Eugenia Rosado María
One important piece of work in the classifications started by the seminal works of S. Lie [16], [17] is the classification of four-dimensional Lorentzian real reductive pairs. This classification appeared, except for one paper, as preprints of the University of Oslo, where moreover many proofs and implications are (necessarily, due to their length) greatly abridged.
Given the relevance of these classifications, we think that an article on the origin, context, methods and relevance of that classification is in order. This is precisely the aim of the present paper. We intend to fill the gaps in the exposition of the ideas that structure these proofs.
On the other hand, motivated by the physical applications, we studied in [3] which of Lorentzian symmetric pairs furnish connected simply-connected Einstein-Yang-Mills spaces, obtaining 10 spaces. Since the calculations are rather long (some one hundred fifty pages, only for these cases), we confine ourselves in the present paper to carefully check the arguments for those 10 cases.
{"title":"On the local classification of four-dimensional Lorentzian real reductive pairs","authors":"Marco Castrillón López , Pedro M. Gadea , Boris P. Komrakov , M. Eugenia Rosado María","doi":"10.1016/j.difgeo.2026.102333","DOIUrl":"10.1016/j.difgeo.2026.102333","url":null,"abstract":"<div><div>One important piece of work in the classifications started by the seminal works of S. Lie <span><span>[16]</span></span>, <span><span>[17]</span></span> is the classification of four-dimensional Lorentzian real reductive pairs. This classification appeared, except for one paper, as preprints of the University of Oslo, where moreover many proofs and implications are (necessarily, due to their length) greatly abridged.</div><div>Given the relevance of these classifications, we think that an article on the origin, context, methods and relevance of that classification is in order. This is precisely the aim of the present paper. We intend to fill the gaps in the exposition of the ideas that structure these proofs.</div><div>On the other hand, motivated by the physical applications, we studied in <span><span>[3]</span></span> which of Lorentzian symmetric pairs furnish connected simply-connected Einstein-Yang-Mills spaces, obtaining 10 spaces. Since the calculations are rather long (some one hundred fifty pages, only for these cases), we confine ourselves in the present paper to carefully check the arguments for those 10 cases.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102333"},"PeriodicalIF":0.7,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981241","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 : 2026-01-13DOI: 10.1016/j.difgeo.2026.102330
Fagui Li , Junrong Yan
Let Σ be a closed, embedded, oriented hypersurface in a closed oriented Riemannian manifold N. Under a lower bound on the Ricci curvature and an upper bound on the sectional curvature of N, we establish a lower bound for the first nonzero eigenvalue of the Laplacian on Σ. The estimate depends on the ambient curvature bounds, the normal injectivity radius, and the geometry of Σ through its mean curvature and second fundamental form. This result extends the classical eigenvalue estimate of Choi and Wang [J. Diff. Geom. 18 (1983), 559–562.] to the non-minimal case.
{"title":"A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds","authors":"Fagui Li , Junrong Yan","doi":"10.1016/j.difgeo.2026.102330","DOIUrl":"10.1016/j.difgeo.2026.102330","url":null,"abstract":"<div><div>Let Σ be a closed, embedded, oriented hypersurface in a closed oriented Riemannian manifold <em>N</em>. Under a lower bound on the Ricci curvature and an upper bound on the sectional curvature of <em>N</em>, we establish a lower bound for the first nonzero eigenvalue of the Laplacian on Σ. The estimate depends on the ambient curvature bounds, the normal injectivity radius, and the geometry of Σ through its mean curvature and second fundamental form. This result extends the classical eigenvalue estimate of Choi and Wang [J. Diff. Geom. <strong>18</strong> (1983), 559–562.] to the non-minimal case.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102330"},"PeriodicalIF":0.7,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976689","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 : 2026-01-13DOI: 10.1016/j.difgeo.2025.102328
A. Mezrag, Z. Muzsnay, Cs. Vincze
In this paper, we consider the geometric setting of navigation data and introduce a natural parallel translation using the Riemannian parallelism. The geometry obtained in this way has some nice and natural features: the natural parallel translation is homogeneous (but in general nonlinear), preserves the Randers type Finslerian norm constituted by the navigation data, and the holonomy group is finite-dimensional.
{"title":"Natural parallel translation and connection associated to navigation data","authors":"A. Mezrag, Z. Muzsnay, Cs. Vincze","doi":"10.1016/j.difgeo.2025.102328","DOIUrl":"10.1016/j.difgeo.2025.102328","url":null,"abstract":"<div><div>In this paper, we consider the geometric setting of navigation data and introduce a natural parallel translation using the Riemannian parallelism. The geometry obtained in this way has some nice and natural features: the natural parallel translation is homogeneous (but in general nonlinear), preserves the Randers type Finslerian norm constituted by the navigation data, and the holonomy group is finite-dimensional.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102328"},"PeriodicalIF":0.7,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976690","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 : 2026-01-12DOI: 10.1016/j.difgeo.2025.102326
Jaume Llibre , Adrian C. Murza
We analyze the polynomial vector fields in that have the hyperboloid of one sheet as an invariant surface. We characterize the maximum number of invariant meridians and parallels of this hyperboloid that such polynomial vector fields can have, as a function of the degree of the polynomial vector field.
{"title":"Invariant meridians and parallels on the hyperboloid of one sheet","authors":"Jaume Llibre , Adrian C. Murza","doi":"10.1016/j.difgeo.2025.102326","DOIUrl":"10.1016/j.difgeo.2025.102326","url":null,"abstract":"<div><div>We analyze the polynomial vector fields in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> that have the hyperboloid of one sheet as an invariant surface. We characterize the maximum number of invariant meridians and parallels of this hyperboloid that such polynomial vector fields can have, as a function of the degree of the polynomial vector field.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102326"},"PeriodicalIF":0.7,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976693","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 : 2026-01-12DOI: 10.1016/j.difgeo.2025.102327
Hakar Raji, Oliver C. Schnürer
There exist rotationally symmetric translating solutions to mean curvature flow that can be written as a graph over Euclidean space. This result is well-known. Its proof uses the symmetry and techniques from partial differential equations. However, the result can also be formulated as an existence result for a singular ordinary differential equation. Here, we provide different methods to prove existence of these solutions based on the study of the singular ordinary differential equation without using methods from partial differential equations.
{"title":"Existence proofs for rotationally symmetric translating solutions to mean curvature flow","authors":"Hakar Raji, Oliver C. Schnürer","doi":"10.1016/j.difgeo.2025.102327","DOIUrl":"10.1016/j.difgeo.2025.102327","url":null,"abstract":"<div><div>There exist rotationally symmetric translating solutions to mean curvature flow that can be written as a graph over Euclidean space. This result is well-known. Its proof uses the symmetry and techniques from partial differential equations. However, the result can also be formulated as an existence result for a singular ordinary differential equation. Here, we provide different methods to prove existence of these solutions based on the study of the singular ordinary differential equation without using methods from partial differential equations.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"102 ","pages":"Article 102327"},"PeriodicalIF":0.7,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976692","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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