Pub Date : 2021-01-01DOI: 10.1515/advgeom-2020-0025
Gianluca Faraco
Abstract Let S be a surface of genus g at least 2. A representation ρ:π1S→PSL2R $ rho:pi_1 Sto{mathrm{PSL}_2mathbb{R}} $ is said to be purely hyperbolic if its image consists only of hyperbolic elements along with the identity. We may wonder under which conditions such representations arise as the holonomy of a branched hyperbolic structure on S. In this work we characterise them completely, giving necessary and sufficient conditions.
{"title":"Geometrisation of purely hyperbolic representations in PSL2R $ {mathrm{PSL}_2mathbb{R}} $","authors":"Gianluca Faraco","doi":"10.1515/advgeom-2020-0025","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0025","url":null,"abstract":"Abstract Let S be a surface of genus g at least 2. A representation ρ:π1S→PSL2R $ rho:pi_1 Sto{mathrm{PSL}_2mathbb{R}} $ is said to be purely hyperbolic if its image consists only of hyperbolic elements along with the identity. We may wonder under which conditions such representations arise as the holonomy of a branched hyperbolic structure on S. In this work we characterise them completely, giving necessary and sufficient conditions.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"21 1","pages":"99 - 108"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2020-0025","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46023560","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 : 2020-12-29DOI: 10.1515/advgeom-2023-0001
Jean-Yves Welschinger
Abstract An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a tiling, encoded by its h-vector, is determined by the number of critical tiles of each index it uses, encoded by its critical vector. In the case of closed triangulated manifolds, these vectors satisfy some palindromic property. We finally study the behavior of tilings under any stellar subdivision.
{"title":"Shellable tilings on relative simplicial complexes and their h-vectors","authors":"Jean-Yves Welschinger","doi":"10.1515/advgeom-2023-0001","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0001","url":null,"abstract":"Abstract An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a tiling, encoded by its h-vector, is determined by the number of critical tiles of each index it uses, encoded by its critical vector. In the case of closed triangulated manifolds, these vectors satisfy some palindromic property. We finally study the behavior of tilings under any stellar subdivision.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"23 1","pages":"191 - 206"},"PeriodicalIF":0.5,"publicationDate":"2020-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41820164","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 : 2020-12-04DOI: 10.1515/advgeom-2022-0004
María Valeria Gutiérrez
Abstract Given a nilpotent Lie algebra, we study the space of all diagonalizable derivations such that the corresponding one-dimensional solvable extension admits a left-invariant metric with negative Ricci curvature. Lauret and Will have conjectured that such a space coincides with an open and convex subset of derivations defined in terms of the moment map for the variety of nilpotent Lie algebras. We prove the validity of the conjecture in dimensions ≤ 5, as well as for Heisenberg Lie algebras and standard filiform Lie algebras.
{"title":"On Ricci negative derivations","authors":"María Valeria Gutiérrez","doi":"10.1515/advgeom-2022-0004","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0004","url":null,"abstract":"Abstract Given a nilpotent Lie algebra, we study the space of all diagonalizable derivations such that the corresponding one-dimensional solvable extension admits a left-invariant metric with negative Ricci curvature. Lauret and Will have conjectured that such a space coincides with an open and convex subset of derivations defined in terms of the moment map for the variety of nilpotent Lie algebras. We prove the validity of the conjecture in dimensions ≤ 5, as well as for Heisenberg Lie algebras and standard filiform Lie algebras.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"22 1","pages":"199 - 214"},"PeriodicalIF":0.5,"publicationDate":"2020-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46454841","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 : 2020-10-22DOI: 10.1515/advgeom-2023-0009
S. Kannan, Shiyue Li, S. Serpente, Claudia He Yun
Abstract We study the topology of moduli spaces of weighted stable tropical curves Δg,w with fixed genus and unit volume. The space Δg,w arises as the dual complex of the divisor of singular curves in Hassett’s moduli space Mg,w of weighted stable curves. When the genus is positive, we show that Δg,w is simply connected for any choice of weight vector w. We also give a formula for the Euler characteristic of Δg,w in terms of the combinatorics of the weight vector.
{"title":"Topology of tropical moduli spaces of weighted stable curves in higher genus","authors":"S. Kannan, Shiyue Li, S. Serpente, Claudia He Yun","doi":"10.1515/advgeom-2023-0009","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0009","url":null,"abstract":"Abstract We study the topology of moduli spaces of weighted stable tropical curves Δg,w with fixed genus and unit volume. The space Δg,w arises as the dual complex of the divisor of singular curves in Hassett’s moduli space Mg,w of weighted stable curves. When the genus is positive, we show that Δg,w is simply connected for any choice of weight vector w. We also give a formula for the Euler characteristic of Δg,w in terms of the combinatorics of the weight vector.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"23 1","pages":"305 - 314"},"PeriodicalIF":0.5,"publicationDate":"2020-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48986446","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 : 2020-10-01DOI: 10.1515/advgeom-2020-frontmatter4
{"title":"Frontmatter","authors":"","doi":"10.1515/advgeom-2020-frontmatter4","DOIUrl":"https://doi.org/10.1515/advgeom-2020-frontmatter4","url":null,"abstract":"","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2020-frontmatter4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45676270","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 : 2020-10-01DOI: 10.1515/advgeom-2019-0023
J. Lang
Abstract It is proven by elementary methods that in dimension 2, every locally injective continuous map, sending the curves of a Ck-spray to curves of another Ck-spray as oriented point sets, is a Ck-diffeomorphism. This extends the result [1] for dimension three and higher from 1965.
{"title":"Differentiability of projective transformations in dimension 2","authors":"J. Lang","doi":"10.1515/advgeom-2019-0023","DOIUrl":"https://doi.org/10.1515/advgeom-2019-0023","url":null,"abstract":"Abstract It is proven by elementary methods that in dimension 2, every locally injective continuous map, sending the curves of a Ck-spray to curves of another Ck-spray as oriented point sets, is a Ck-diffeomorphism. This extends the result [1] for dimension three and higher from 1965.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"20 1","pages":"553 - 557"},"PeriodicalIF":0.5,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2019-0023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46929442","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 : 2020-10-01DOI: 10.1515/advgeom-2020-0003
Á. L. M. Castañeda
Abstract We prove the existence of a linearization for singular principal G-bundles not depending on the base curve. This allow us to construct the relative compact moduli space of δ-(semi)stable singular principal G-bundles over families of reduced projective and connected nodal curves, and to reduce the construction of the universal moduli space over 𝓜g to the construction of the universal moduli space of swamps.
{"title":"On the moduli spaces of singular principal bundles on stable curves","authors":"Á. L. M. Castañeda","doi":"10.1515/advgeom-2020-0003","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0003","url":null,"abstract":"Abstract We prove the existence of a linearization for singular principal G-bundles not depending on the base curve. This allow us to construct the relative compact moduli space of δ-(semi)stable singular principal G-bundles over families of reduced projective and connected nodal curves, and to reduce the construction of the universal moduli space over 𝓜g to the construction of the universal moduli space of swamps.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"20 1","pages":"573 - 584"},"PeriodicalIF":0.5,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2020-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47276369","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 : 2020-09-22DOI: 10.1515/ADVGEOM-2020-0015
Shaoxiang Zhang, Zaili Yan
Abstract Studying geodesic orbit Randers metrics on spheres, we obtain a complete classification of such metrics. Our method relies upon the classification of geodesic orbit Riemannian metrics on the spheres Sn in [17] and the navigation data in Finsler geometry. We also construct some explicit U(n + 1)-invariant metrics on S2n+1 and Sp(n + 1)U(1)-invariant metrics on S4n+3.
{"title":"Geodesic orbit Randers metrics on spheres","authors":"Shaoxiang Zhang, Zaili Yan","doi":"10.1515/ADVGEOM-2020-0015","DOIUrl":"https://doi.org/10.1515/ADVGEOM-2020-0015","url":null,"abstract":"Abstract Studying geodesic orbit Randers metrics on spheres, we obtain a complete classification of such metrics. Our method relies upon the classification of geodesic orbit Riemannian metrics on the spheres Sn in [17] and the navigation data in Finsler geometry. We also construct some explicit U(n + 1)-invariant metrics on S2n+1 and Sp(n + 1)U(1)-invariant metrics on S4n+3.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"21 1","pages":"273 - 280"},"PeriodicalIF":0.5,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43885027","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 : 2020-09-01DOI: 10.1515/advgeom-2021-0036
Alice Lim
Abstract In this paper, we classify the compact locally homogeneous non-gradient m-quasi Einstein 3- manifolds. Along the way, we also prove that given a compact quotient of a Lie group of any dimension that is m-quasi Einstein, the potential vector field X must be left invariant and Killing. We also classify the nontrivial m-quasi Einstein metrics that are a compact quotient of the product of two Einstein metrics. We also show that S1 is the only compact manifold of any dimension which admits a metric which is nontrivially m-quasi Einstein and Einstein.
{"title":"Locally homogeneous non-gradient quasi-Einstein 3-manifolds","authors":"Alice Lim","doi":"10.1515/advgeom-2021-0036","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0036","url":null,"abstract":"Abstract In this paper, we classify the compact locally homogeneous non-gradient m-quasi Einstein 3- manifolds. Along the way, we also prove that given a compact quotient of a Lie group of any dimension that is m-quasi Einstein, the potential vector field X must be left invariant and Killing. We also classify the nontrivial m-quasi Einstein metrics that are a compact quotient of the product of two Einstein metrics. We also show that S1 is the only compact manifold of any dimension which admits a metric which is nontrivially m-quasi Einstein and Einstein.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"22 1","pages":"79 - 93"},"PeriodicalIF":0.5,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48621254","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 : 2020-08-19DOI: 10.1515/advgeom-2021-0028
Oliver Goertsches, E. Loiudice
Abstract We show that any compact metric f-K-contact, respectively S-manifold is obtained from a compact K-contact, respectively Sasakian manifold by an iteration of constructions of mapping tori, rotations, and type II deformations.
{"title":"How to construct all metric f-K-contact manifolds","authors":"Oliver Goertsches, E. Loiudice","doi":"10.1515/advgeom-2021-0028","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0028","url":null,"abstract":"Abstract We show that any compact metric f-K-contact, respectively S-manifold is obtained from a compact K-contact, respectively Sasakian manifold by an iteration of constructions of mapping tori, rotations, and type II deformations.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"21 1","pages":"591 - 598"},"PeriodicalIF":0.5,"publicationDate":"2020-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49312824","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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