{"title":"Induced almost para-Kähler Einstein metrics on cotangent bundles","authors":"Andreas Čap, Thomas Mettler","doi":"10.1093/qmath/haae047","DOIUrl":null,"url":null,"abstract":"In earlier work, we have shown that for certain geometric structures on a smooth manifold M of dimension n, one obtains an almost para-Kähler–Einstein metric on a manifold A of dimension 2n associated to the structure on M. The geometry also associates a diffeomorphism between A and $T^*M$ to any torsion-free connection compatible with the geometric structure. Hence we can use this construction to associate to each compatible connection an almost para-Kähler–Einstein metric on $T^*M$. In this short article, we discuss the relation of these metrics to Patterson–Walker metrics and derive explicit formulae for them in the cases of projective, conformal and Grassmannian structures.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"12 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quarterly Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/qmath/haae047","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In earlier work, we have shown that for certain geometric structures on a smooth manifold M of dimension n, one obtains an almost para-Kähler–Einstein metric on a manifold A of dimension 2n associated to the structure on M. The geometry also associates a diffeomorphism between A and $T^*M$ to any torsion-free connection compatible with the geometric structure. Hence we can use this construction to associate to each compatible connection an almost para-Kähler–Einstein metric on $T^*M$. In this short article, we discuss the relation of these metrics to Patterson–Walker metrics and derive explicit formulae for them in the cases of projective, conformal and Grassmannian structures.
在早先的工作中,我们已经证明,对于维数为 n 的光滑流形 M 上的某些几何结构,我们可以在维数为 2n 的流形 A 上得到一个与 M 上的结构相关联的近似对凯勒-爱因斯坦度量。因此,我们可以利用这一构造,在 $T^*M$ 上为每个兼容连接关联一个几乎准凯勒-爱因斯坦度量。在这篇短文中,我们将讨论这些度量与帕特森-沃克度量的关系,并推导出它们在投影结构、共形结构和格拉斯曼结构情况下的明确公式。
期刊介绍:
The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.