{"title":"A geometric splitting theorem for actions of semisimple Lie groups","authors":"José Rosales-Ortega","doi":"10.1007/s12188-021-00242-2","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>M</i> be a compact connected smooth pseudo-Riemannian manifold that admits a topologically transitive <i>G</i>-action by isometries, where <span>\\(G = G_1 \\ldots G_l\\)</span> is a connected semisimple Lie group without compact factors whose Lie algebra is <span>\\({\\mathfrak {g}}= {\\mathfrak {g}}_1 \\oplus {\\mathfrak {g}}_2 \\oplus \\cdots \\oplus {\\mathfrak {g}}_l\\)</span>. If <span>\\(m_0,n_0,n_0^i\\)</span> are the dimensions of the maximal lightlike subspaces tangent to <i>M</i>, <i>G</i>, <span>\\(G_i\\)</span>, respectively, then we study <i>G</i>-actions that satisfy the condition <span>\\(m_0=n_0^1 + \\cdots + n_0^{l}\\)</span>. This condition implies that the orbits are non-degenerate for the pseudo Riemannian metric on <i>M</i> and this allows us to consider the normal bundle to the orbits. Using the properties of the normal bundle to the <i>G</i>-orbits we obtain an isometric splitting of <i>M</i> by considering natural metrics on each <span>\\(G_i\\)</span>.</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2021-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12188-021-00242-2","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s12188-021-00242-2","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let M be a compact connected smooth pseudo-Riemannian manifold that admits a topologically transitive G-action by isometries, where \(G = G_1 \ldots G_l\) is a connected semisimple Lie group without compact factors whose Lie algebra is \({\mathfrak {g}}= {\mathfrak {g}}_1 \oplus {\mathfrak {g}}_2 \oplus \cdots \oplus {\mathfrak {g}}_l\). If \(m_0,n_0,n_0^i\) are the dimensions of the maximal lightlike subspaces tangent to M, G, \(G_i\), respectively, then we study G-actions that satisfy the condition \(m_0=n_0^1 + \cdots + n_0^{l}\). This condition implies that the orbits are non-degenerate for the pseudo Riemannian metric on M and this allows us to consider the normal bundle to the orbits. Using the properties of the normal bundle to the G-orbits we obtain an isometric splitting of M by considering natural metrics on each \(G_i\).
期刊介绍:
The first issue of the "Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg" was published in the year 1921. This international mathematical journal has since then provided a forum for significant research contributions. The journal covers all central areas of pure mathematics, such as algebra, complex analysis and geometry, differential geometry and global analysis, graph theory and discrete mathematics, Lie theory, number theory, and algebraic topology.