A geometric splitting theorem for actions of semisimple Lie groups

IF 0.4 4区 数学 Q4 MATHEMATICS
José Rosales-Ortega
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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 \(G_i\).

半单李群作用的几何分裂定理
设M为一个紧连通光滑伪黎曼流形,其具有拓扑可传递g作用,其中\(G = G_1 \ldots G_l\)为一个连通的不紧因子的半单李群,其李代数为\({\mathfrak {g}}= {\mathfrak {g}}_1 \oplus {\mathfrak {g}}_2 \oplus \cdots \oplus {\mathfrak {g}}_l\)。如果\(m_0,n_0,n_0^i\)分别是与M, G, \(G_i\)相切的最大类光子空间的维数,那么我们研究满足\(m_0=n_0^1 + \cdots + n_0^{l}\)条件的G作用。这个条件意味着M上伪黎曼度规的轨道是非简并的这允许我们考虑轨道的正常束。利用g轨道的法向束的性质,我们通过考虑每个\(G_i\)上的自然度量获得M的等距分裂。
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来源期刊
CiteScore
0.80
自引率
0.00%
发文量
7
审稿时长
>12 weeks
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.
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