{"title":"Metrics of positive Ricci curvature on simply-connected manifolds of dimension \n \n \n 6\n k\n \n $6k$","authors":"Philipp Reiser","doi":"10.1112/topo.70007","DOIUrl":null,"url":null,"abstract":"<p>A consequence of the surgery theorem of Gromov and Lawson is that every closed, simply-connected 6-manifold admits a Riemannian metric of positive scalar curvature. For metrics of positive Ricci curvature, it is widely open whether a similar result holds; there are no obstructions known for those manifolds to admit a metric of positive Ricci curvature, while the number of examples known is limited. In this article, we introduce a new description of certain <span></span><math>\n <semantics>\n <mrow>\n <mn>6</mn>\n <mi>k</mi>\n </mrow>\n <annotation>$6k$</annotation>\n </semantics></math>-dimensional manifolds via labeled bipartite graphs and use an earlier result of the author to construct metrics of positive Ricci curvature on these manifolds. In this way, we obtain many new examples, both spin and nonspin, of <span></span><math>\n <semantics>\n <mrow>\n <mn>6</mn>\n <mi>k</mi>\n </mrow>\n <annotation>$6k$</annotation>\n </semantics></math>-dimensional manifolds with a metric of positive Ricci curvature.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 4","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.70007","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Topology","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/topo.70007","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A consequence of the surgery theorem of Gromov and Lawson is that every closed, simply-connected 6-manifold admits a Riemannian metric of positive scalar curvature. For metrics of positive Ricci curvature, it is widely open whether a similar result holds; there are no obstructions known for those manifolds to admit a metric of positive Ricci curvature, while the number of examples known is limited. In this article, we introduce a new description of certain -dimensional manifolds via labeled bipartite graphs and use an earlier result of the author to construct metrics of positive Ricci curvature on these manifolds. In this way, we obtain many new examples, both spin and nonspin, of -dimensional manifolds with a metric of positive Ricci curvature.
格罗莫夫（Gromov）和劳森（Lawson）的手术定理的一个结果是，每一个封闭的、简单连接的 6-manifold（6-manifold）都容许一个具有正标度曲率的黎曼度量。对于正利玛窦曲率的度量，类似的结果是否成立还没有定论；目前还不知道这些流形是否存在接纳正利玛窦曲率度量的障碍，而已知的例子数量有限。在这篇文章中，我们通过带标签的二叉图引入了对某些 6 k $6k$ 维流形的新描述，并利用作者早期的一个结果在这些流形上构造了正利玛窦曲率度量。通过这种方法，我们得到了许多具有正利玛窦曲率度量的 6 k $6k$ -维流形的新例子，包括自旋和非自旋流形。
期刊介绍：
The Journal of Topology publishes papers of high quality and significance in topology, geometry and adjacent areas of mathematics. Interesting, important and often unexpected links connect topology and geometry with many other parts of mathematics, and the editors welcome submissions on exciting new advances concerning such links, as well as those in the core subject areas of the journal.
The Journal of Topology was founded in 2008. It is published quarterly with articles published individually online prior to appearing in a printed issue.
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