An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures
An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures
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DOI：
10.1090/s0002-9904-1975-13649-4
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发表时间：
1975
影响因子：
1.3
通讯作者：
Simon Aloff;N. Wallach
中科院分区：
文献类型：
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作者：
Simon Aloff;N. Wallach
1.导论.本文的目的是证明：如果T1是SU（3）的一个闭的、连通的、没有非零不动点的一维子群，则SUtyjT 1有一个严格正曲率的SC/^-不变黎曼结构。这个结果暗示了标题的陈述，因为在具有上述T1的空间SUtyjT 1中有无限多个不同的同伦类型（见引理3.3）。为了证明上述结果，我们引入我们称之为条件II的条件，它推广了[2]和[3]的条件III。虽然空间SUty/T1是唯一满足条件II的新空间（见下面的定理5.1），但引入这个概念是值得的，因为Berger [1]的13维例子M2以非平凡的方式满足条件II。因此，在M2上存在一组不变度量<，）t，其中-l < f<ε是严格正曲率的（见§ 4），（，）0是唯一的正规度量。本文的主要结果是由作者用稍微不同的方法独立发现的。我们已经给出了第二作者的技术，因为它在概念上更简单。引理3.3是由芝加哥大学的Lashof教授向第二作者指出的，并由第一作者独立推导出来的。
1. Introduction. The purpose of this note is to show that if T1 is a closed, connected one-dimensional subgroup of SU (3) that has no nonzero fixed points, then SUtyjT1 admits an SC/^-invariant Riemannian structure of strictly positive curvature. This result implies the statement of the title since there are an infinite number of distinct homotopy types among the spaces SUtyjT1 with T1 as above (see Lemma 3.3). To prove the above result we introduce what we call condition II, which generalizes condition III of [2] and [3]. Although the spaces SUty/T1 are the only new spaces satisfying condition II (see Theorem 5.1 below), it is worthwhile to introduce the notion since the 13-dimensional example, M2, of Berger [1] satisfies condition II in a nontrivial fashion. Hence there is a set of invariant metrics<,) t on M2 with—l< f<£ of strictly positive curvature (see § 4) with (,) 0 the only normal one.A word should be said about the joint authorship of this paper. The main result of this note was found independently by the authors by slightly different methods. We have given the technique of the second author since it is conceptually simpler. Lemma 3.3 was pointed out to the second author by Professor Lashof of the University of Chicago and derived independently by the first author.