Alice Le Brigant, Leandro Lichtenfelz, Stephen C. Preston
In a Lie group equipped with a left-invariant metric, we study the minimizing properties of geodesics through the presence of conjugate points. We give criteria for the existence of conjugate points along steady and nonsteady geodesics, using different strategies in each case. We consider both general Lie groups and quadratic Lie groups, where the metric in the Lie algebra $g(u,v)=langle u,Lambda vrangle$ is defined from a bi-invariant bilinear form and a symmetric positive definite operator $Lambda$. By way of illustration, we apply our criteria to $SO(n)$ equipped with a generalized version of the rigid body metric, and to Lie groups arising from Cheeger's deformation technique, which include Zeitlin's $SU(3)$ model of hydrodynamics on the $2$-sphere. Along the way we obtain formulas for the Ricci curvatures in these examples, showing that conjugate points occur even in the presence of some negative curvature.
{"title":"Geodesics, curvature, and conjugate points on Lie groups","authors":"Alice Le Brigant, Leandro Lichtenfelz, Stephen C. Preston","doi":"arxiv-2408.03854","DOIUrl":"https://doi.org/arxiv-2408.03854","url":null,"abstract":"In a Lie group equipped with a left-invariant metric, we study the minimizing\u0000properties of geodesics through the presence of conjugate points. We give\u0000criteria for the existence of conjugate points along steady and nonsteady\u0000geodesics, using different strategies in each case. We consider both general\u0000Lie groups and quadratic Lie groups, where the metric in the Lie algebra\u0000$g(u,v)=langle u,Lambda vrangle$ is defined from a bi-invariant bilinear\u0000form and a symmetric positive definite operator $Lambda$. By way of\u0000illustration, we apply our criteria to $SO(n)$ equipped with a generalized\u0000version of the rigid body metric, and to Lie groups arising from Cheeger's\u0000deformation technique, which include Zeitlin's $SU(3)$ model of hydrodynamics\u0000on the $2$-sphere. Along the way we obtain formulas for the Ricci curvatures in\u0000these examples, showing that conjugate points occur even in the presence of\u0000some negative curvature.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Felippe Guimarães, Fernando Manfio, Carlos E. Olmos
We study isometric immersions $f: M^n rightarrow mathbb{H}^{n+1}$ into hyperbolic space of dimension $n+1$ of a complete Riemannian manifold of dimension $n$ on which a compact connected group of intrinsic isometries acts with principal orbits of codimension one. We provide a characterization if either $n geq 3$ and $M^n$ is compact, or $n geq 5$ and the connected components of the set where the sectional curvature is constant and equal to $-1$ are bounded.
{"title":"Complete cohomogeneity one hypersurfaces of $mathbb{H}^{n+1}$","authors":"Felippe Guimarães, Fernando Manfio, Carlos E. Olmos","doi":"arxiv-2408.03802","DOIUrl":"https://doi.org/arxiv-2408.03802","url":null,"abstract":"We study isometric immersions $f: M^n rightarrow mathbb{H}^{n+1}$ into\u0000hyperbolic space of dimension $n+1$ of a complete Riemannian manifold of\u0000dimension $n$ on which a compact connected group of intrinsic isometries acts\u0000with principal orbits of codimension one. We provide a characterization if\u0000either $n geq 3$ and $M^n$ is compact, or $n geq 5$ and the connected\u0000components of the set where the sectional curvature is constant and equal to\u0000$-1$ are bounded.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a lower bound for the delta invariant of the fundamental divisor of a quasi-smooth weighted hypersurface. As a consequence, we prove K-stability of a large class of quasi-smooth Fano hypersurfaces of index 1 and of all smooth Fano weighted hypersurfaces of index 1 and 2. The proofs are based on the Abban--Zhuang method and on the study of linear systems on flags of weighted hypersurfaces.
我们给出了准光滑加权超曲面基底除数的三角不变量下限。因此，我们证明了指数为 1 的一大类准光滑法诺超曲面以及指数为 1 和 2 的所有光滑法诺加权超曲面的 K 稳定性。证明基于阿班--庄方法和加权超曲面旗上线性系统的研究。
{"title":"Delta invariants of weighted hypersurfaces","authors":"Taro Sano, Luca Tasin","doi":"arxiv-2408.03057","DOIUrl":"https://doi.org/arxiv-2408.03057","url":null,"abstract":"We give a lower bound for the delta invariant of the fundamental divisor of a\u0000quasi-smooth weighted hypersurface. As a consequence, we prove K-stability of a\u0000large class of quasi-smooth Fano hypersurfaces of index 1 and of all smooth\u0000Fano weighted hypersurfaces of index 1 and 2. The proofs are based on the\u0000Abban--Zhuang method and on the study of linear systems on flags of weighted\u0000hypersurfaces.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"113 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a blow-up formula for Bott-Chern characteristic classes of compact complex manifolds. To this end, we establish a version of Riemann-Roch without denominators for the Bott-Chern characteristic classes. In particular, as an application, we study the behaviour of the Bott-Chern characteristic classes of the Iwasawa manifold under a blow-up transformation.
{"title":"Bott-Chern characteristic classes of blow-ups","authors":"Xiaojun Wu, Song Yang, Xiangdong Yang","doi":"arxiv-2408.03210","DOIUrl":"https://doi.org/arxiv-2408.03210","url":null,"abstract":"We prove a blow-up formula for Bott-Chern characteristic classes of compact\u0000complex manifolds. To this end, we establish a version of Riemann-Roch without\u0000denominators for the Bott-Chern characteristic classes. In particular, as an\u0000application, we study the behaviour of the Bott-Chern characteristic classes of\u0000the Iwasawa manifold under a blow-up transformation.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"161 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain the convergence speed for Fekete points on uniformly polynomially cuspidal compact sets introduced by Pawlucki and Ple'sniak. This is done by showing that these sets are $(mathscr{C}^{alpha}, mathscr{C}^{alpha'})$-regular in the sense of Dinh, Ma and Nguyen.
我们得到了 Pawlucki 和 Ple'sniak 引入的均匀多项式cuspidal 紧凑集上的 Fekete 点的收敛速度。这是通过证明这些集合在丁（Dinh）、马（Ma）和阮（Nguyen）的意义上是$(mathscr{C}^{alpha},mathscr{C}^{alpha'})$正则集合来实现的。
{"title":"Convergence Speed for Fekete Points on Uniformly Polynomially Cuspidal Sets","authors":"Hyunsoo Ahn, Ngoc Cuong Nguyen","doi":"arxiv-2408.03053","DOIUrl":"https://doi.org/arxiv-2408.03053","url":null,"abstract":"We obtain the convergence speed for Fekete points on uniformly polynomially\u0000cuspidal compact sets introduced by Pawlucki and Ple'sniak. This is done by\u0000showing that these sets are $(mathscr{C}^{alpha},\u0000mathscr{C}^{alpha'})$-regular in the sense of Dinh, Ma and Nguyen.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we compute the limit of the Wang--Yau quasi-local mass on a family of surfaces approaching the apparent horizon (the near horizon limit). Such limit is first considered in [1]. Recently, Pook-Kolb, Zhao, Andersson, Krishnan, and Yau investigated the near horizon limit of the Wang--Yau quasi-local mass in binary black hole mergers in [12] and conjectured that the optimal embeddings approach the isometric embedding of the horizon into $R^3$. Moreover, the quasi-local mass converges to the total mean curvature of the image. The vanishing of the norm of the mean curvature vector implies special properties for the Wang--Yau quasi-local energy and the optimal embedding equation. We utilize these features to prove the existence and uniqueness of the optimal embedding and investigate the minimization of the Wang--Yau quasi-local energy. In particular, we prove the continuity of the quasi-local mass in the near horizon limit.
{"title":"Near horizon limit of the Wang--Yau quasi-local mass","authors":"Po-Ning Chen","doi":"arxiv-2408.02917","DOIUrl":"https://doi.org/arxiv-2408.02917","url":null,"abstract":"In this article, we compute the limit of the Wang--Yau quasi-local mass on a\u0000family of surfaces approaching the apparent horizon (the near horizon limit).\u0000Such limit is first considered in [1]. Recently, Pook-Kolb, Zhao, Andersson,\u0000Krishnan, and Yau investigated the near horizon limit of the Wang--Yau\u0000quasi-local mass in binary black hole mergers in [12] and conjectured that the\u0000optimal embeddings approach the isometric embedding of the horizon into $R^3$.\u0000Moreover, the quasi-local mass converges to the total mean curvature of the\u0000image. The vanishing of the norm of the mean curvature vector implies special\u0000properties for the Wang--Yau quasi-local energy and the optimal embedding\u0000equation. We utilize these features to prove the existence and uniqueness of\u0000the optimal embedding and investigate the minimization of the Wang--Yau\u0000quasi-local energy. In particular, we prove the continuity of the quasi-local\u0000mass in the near horizon limit.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is an expository note to give a brief review of classical elastica theory, mainly prepared for giving a more detailed proof of the author's Li--Yau type inequality for self-intersecting curves in Euclidean space. We also discuss some open problems in related topics.
{"title":"Elastic curves and self-intersections","authors":"Tatsuya Miura","doi":"arxiv-2408.03020","DOIUrl":"https://doi.org/arxiv-2408.03020","url":null,"abstract":"This is an expository note to give a brief review of classical elastica\u0000theory, mainly prepared for giving a more detailed proof of the author's\u0000Li--Yau type inequality for self-intersecting curves in Euclidean space. We\u0000also discuss some open problems in related topics.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The notion of a emph{higher-order algebroid}, as introduced in cite{MJ_MR_HA_comorph_2018}, generalizes the concepts of a higher-order tangent bundle $tau^k_M: mathrm{T}^k M rightarrow M$ and a (Lie) algebroid. This idea is based on a (vector bundle) comorphism approach to (Lie) algebroids and the reduction procedure of homotopies from the level of Lie groupoids to that of Lie algebroids. In brief, an alternative description of a Lie algebroid $(A, [cdot, cdot], sharp)$ is a vector bundle comorphism $kappa$ defined as the dual of the Poisson map $varepsilon: mathrm{T}^ast A rightarrow mathrm{T} A^ast$ associated with the Lie algebroid $A$. The framework of comorphisms has proven to be a suitable language for describing higher-order analogues of Lie algebroids from the perspective of the role played by (Lie) algebroids in geometric mechanics. In this work, we uncover the classical algebraic structures underlying the mysterious description of higher-order algebroids through comorphisms. For the case where $k=2$, we establish one-to-one correspondence between higher-order Lie algebroids and pairs consisting of a two-term representation (up to homotopy) of a Lie algebroid and a morphism to the adjoint representation of this algebroid.
{"title":"Exploring the Structure of Higher Algebroids","authors":"Mikołaj Rotkiewicz","doi":"arxiv-2408.02194","DOIUrl":"https://doi.org/arxiv-2408.02194","url":null,"abstract":"The notion of a emph{higher-order algebroid}, as introduced in\u0000cite{MJ_MR_HA_comorph_2018}, generalizes the concepts of a higher-order\u0000tangent bundle $tau^k_M: mathrm{T}^k M rightarrow M$ and a (Lie) algebroid.\u0000This idea is based on a (vector bundle) comorphism approach to (Lie) algebroids\u0000and the reduction procedure of homotopies from the level of Lie groupoids to\u0000that of Lie algebroids. In brief, an alternative description of a Lie algebroid\u0000$(A, [cdot, cdot], sharp)$ is a vector bundle comorphism $kappa$ defined as\u0000the dual of the Poisson map $varepsilon: mathrm{T}^ast A rightarrow\u0000mathrm{T} A^ast$ associated with the Lie algebroid $A$. The framework of\u0000comorphisms has proven to be a suitable language for describing higher-order\u0000analogues of Lie algebroids from the perspective of the role played by (Lie)\u0000algebroids in geometric mechanics. In this work, we uncover the classical\u0000algebraic structures underlying the mysterious description of higher-order\u0000algebroids through comorphisms. For the case where $k=2$, we establish\u0000one-to-one correspondence between higher-order Lie algebroids and pairs\u0000consisting of a two-term representation (up to homotopy) of a Lie algebroid and\u0000a morphism to the adjoint representation of this algebroid.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bang-Yen Chen, Majid Ali Choudhary, Afshan Perween
In differential geometry, the concept of golden structure, initially proposed by S. I. Goldberg and K. Yano in 1970, presents a compelling area with wide-ranging applications. The exploration of golden Riemannian manifolds was initiated by C. E. Hretcanu and M. Crasmareanu in 2008, following the principles of the golden structure. Subsequently, numerous researchers have contributed significant insights into golden Riemannian manifolds. The purpose of this paper is to provide a comprehensive survey on golden Riemannian manifold done over the past decade.
在微分几何学中，黄金结构的概念最初是由 S. I. Goldberg 和 K. Yano 于 1970 年提出的，它是一个具有广泛应用的引人注目的领域。2008 年，C. E. Hretcanu 和 M. Crasmareanu 根据黄金结构的原理，开始了对黄金黎曼流形的探索。随后，众多研究人员对黄金黎曼流形发表了重要见解。本文旨在对过去十年间有关黄金黎曼流形的研究进行全面梳理。
{"title":"A comprehensive review of golden Riemannian manifolds","authors":"Bang-Yen Chen, Majid Ali Choudhary, Afshan Perween","doi":"arxiv-2408.02800","DOIUrl":"https://doi.org/arxiv-2408.02800","url":null,"abstract":"In differential geometry, the concept of golden structure, initially proposed\u0000by S. I. Goldberg and K. Yano in 1970, presents a compelling area with\u0000wide-ranging applications. The exploration of golden Riemannian manifolds was\u0000initiated by C. E. Hretcanu and M. Crasmareanu in 2008, following the\u0000principles of the golden structure. Subsequently, numerous researchers have\u0000contributed significant insights into golden Riemannian manifolds. The purpose\u0000of this paper is to provide a comprehensive survey on golden Riemannian\u0000manifold done over the past decade.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Utilizing the covariant formulation of Penrose's plane wave limit by Blau et al., we construct for any Riemannian metric $g$ a family of "plane wave limits" of one higher dimension. These limits are taken along geodesics of $g$, yield simpler metrics of Lorentzian signature, and are isometric invariants. They can also be seen to arise locally from a suitable expansion of $g$ in Fermi coordinates, and they directly encode much of $g$'s geometry. For example, normal Jacobi fields of $g$ are encoded as geodesics of its plane wave limits. Furthermore, $g$ will have constant sectional curvature if and only if each of its plane wave limits is locally conformally flat. In fact $g$ will be flat, or Ricci-flat, or geodesically complete, if and only if all of its plane wave limits are, respectively, the same. Many other curvature properties are preserved in the limit, including certain inequalities, such as signed Ricci curvature.
{"title":"Plane wave limits of Riemannian manifolds","authors":"Amir Babak Aazami","doi":"arxiv-2408.02567","DOIUrl":"https://doi.org/arxiv-2408.02567","url":null,"abstract":"Utilizing the covariant formulation of Penrose's plane wave limit by Blau et\u0000al., we construct for any Riemannian metric $g$ a family of \"plane wave limits\"\u0000of one higher dimension. These limits are taken along geodesics of $g$, yield\u0000simpler metrics of Lorentzian signature, and are isometric invariants. They can\u0000also be seen to arise locally from a suitable expansion of $g$ in Fermi\u0000coordinates, and they directly encode much of $g$'s geometry. For example,\u0000normal Jacobi fields of $g$ are encoded as geodesics of its plane wave limits.\u0000Furthermore, $g$ will have constant sectional curvature if and only if each of\u0000its plane wave limits is locally conformally flat. In fact $g$ will be flat, or\u0000Ricci-flat, or geodesically complete, if and only if all of its plane wave\u0000limits are, respectively, the same. Many other curvature properties are\u0000preserved in the limit, including certain inequalities, such as signed Ricci\u0000curvature.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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