{"title":"联系 GRA Solitons 和广义相对论的应用","authors":"Sourav Nayak, Dhriti Sundar Patra","doi":"10.1007/s00009-024-02703-3","DOIUrl":null,"url":null,"abstract":"<p>This article investigates generalized Ricci almost solitons, also known as GRA solitons, on contact metric manifolds, including the gradient case. At first, we establish that a complete <i>K</i>-contact or Sasakian manifold endowed with a closed GRA soliton satisfying <span>\\(4c_1c_2 \\ne 1\\)</span> is compact Einstein with scalar curvature <span>\\(2n(2n+1)\\)</span>. As for the gradient case, it exhibits an isometry to the unit sphere <span>\\({\\mathbb {S}}^{2n+1}\\)</span>. Subsequently, we identify a few adequate conditions under which a non-trivial complete <i>K</i>-contact manifold with a GRA soliton is trivial (<span>\\(\\eta \\)</span>-Einstein). Following that, we establish certain results on <i>H</i>-contact and complete contact manifolds. We also demonstrate that a non-Sasakian <span>\\((k,\\mu )\\)</span>-contact manifold with a closed GRA soliton is flat for dimension 3, and for higher dimensions, it is locally isometric to the trivial bundle <span>\\({\\mathbb {R}}^{n+1} \\times {\\mathbb {S}}^n(4)\\)</span>, provided <span>\\(4c_1c_2 (1-2n)\\ne 1\\)</span> and <span>\\(c_2\\ne 0\\)</span>. Finally, we discuss a few applications of GRA solitons in general relativity. These include characterizing PF spacetimes with a concircular velocity vector field and determining a sufficient condition for a GRW spacetime to be a PF spacetime.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Contact GRA Solitons and Applications to General Relativity\",\"authors\":\"Sourav Nayak, Dhriti Sundar Patra\",\"doi\":\"10.1007/s00009-024-02703-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This article investigates generalized Ricci almost solitons, also known as GRA solitons, on contact metric manifolds, including the gradient case. At first, we establish that a complete <i>K</i>-contact or Sasakian manifold endowed with a closed GRA soliton satisfying <span>\\\\(4c_1c_2 \\\\ne 1\\\\)</span> is compact Einstein with scalar curvature <span>\\\\(2n(2n+1)\\\\)</span>. As for the gradient case, it exhibits an isometry to the unit sphere <span>\\\\({\\\\mathbb {S}}^{2n+1}\\\\)</span>. Subsequently, we identify a few adequate conditions under which a non-trivial complete <i>K</i>-contact manifold with a GRA soliton is trivial (<span>\\\\(\\\\eta \\\\)</span>-Einstein). Following that, we establish certain results on <i>H</i>-contact and complete contact manifolds. We also demonstrate that a non-Sasakian <span>\\\\((k,\\\\mu )\\\\)</span>-contact manifold with a closed GRA soliton is flat for dimension 3, and for higher dimensions, it is locally isometric to the trivial bundle <span>\\\\({\\\\mathbb {R}}^{n+1} \\\\times {\\\\mathbb {S}}^n(4)\\\\)</span>, provided <span>\\\\(4c_1c_2 (1-2n)\\\\ne 1\\\\)</span> and <span>\\\\(c_2\\\\ne 0\\\\)</span>. Finally, we discuss a few applications of GRA solitons in general relativity. These include characterizing PF spacetimes with a concircular velocity vector field and determining a sufficient condition for a GRW spacetime to be a PF spacetime.</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-07-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00009-024-02703-3\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00009-024-02703-3","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
摘要
本文研究了接触元流形上的广义利玛窦孤子(也称 GRA 孤子),包括梯度情况。首先,我们建立了一个完整的K接触流形或萨萨流形,其封闭的GRA孤子满足\(4c_1c_2 \ne 1\) 是具有标量曲率\(2n(2n+1)\)的紧凑爱因斯坦流形。至于梯度情况,它表现出与单位球的等轴性({\mathbb {S}}^{2n+1}\ )。随后,我们确定了一些适当的条件,在这些条件下,具有 GRA 孤子的非琐碎完整 K-contact 流形是琐碎的((\eta \)-爱因斯坦)。随后,我们建立了关于H接触流形和完全接触流形的某些结果。我们还证明了具有封闭 GRA 孤子的非萨萨基((k,\mu ))-接触流形在维度 3 是平坦的,而对于更高维,它与琐细束 \({\mathbb {R}}^{n+1} 是局部等距的。\times {\mathbb {S}}^n(4)\), provided \(4c_1c_2 (1-2n)\ne 1\) and\(c_2\ne 0\).最后,我们讨论了 GRA 孤子在广义相对论中的一些应用。这些应用包括描述具有协圆速度矢量场的PF时空,以及确定GRW时空成为PF时空的充分条件。
Contact GRA Solitons and Applications to General Relativity
This article investigates generalized Ricci almost solitons, also known as GRA solitons, on contact metric manifolds, including the gradient case. At first, we establish that a complete K-contact or Sasakian manifold endowed with a closed GRA soliton satisfying \(4c_1c_2 \ne 1\) is compact Einstein with scalar curvature \(2n(2n+1)\). As for the gradient case, it exhibits an isometry to the unit sphere \({\mathbb {S}}^{2n+1}\). Subsequently, we identify a few adequate conditions under which a non-trivial complete K-contact manifold with a GRA soliton is trivial (\(\eta \)-Einstein). Following that, we establish certain results on H-contact and complete contact manifolds. We also demonstrate that a non-Sasakian \((k,\mu )\)-contact manifold with a closed GRA soliton is flat for dimension 3, and for higher dimensions, it is locally isometric to the trivial bundle \({\mathbb {R}}^{n+1} \times {\mathbb {S}}^n(4)\), provided \(4c_1c_2 (1-2n)\ne 1\) and \(c_2\ne 0\). Finally, we discuss a few applications of GRA solitons in general relativity. These include characterizing PF spacetimes with a concircular velocity vector field and determining a sufficient condition for a GRW spacetime to be a PF spacetime.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.