Alice Le Brigant, Leandro Lichtenfelz, Stephen C. Preston
{"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
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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
{"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}
{"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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