Isometry groups of simply connected manifolds of nonpositive curvature II
Isometry groups of simply connected manifolds of nonpositive curvature II
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非正曲率单连通流形的等距群 II
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发表时间：
1982
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通讯作者：
P. Eberlein
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作者：
P. Eberlein
设H表示具有非正截面曲率的完备单连通黎曼流形，I（H）表示H的等距群。本文考虑满足对偶条件（定义如下）的子群DI（H）的密度性质，这些密度性质也给出了非紧型黎曼对称空间的特征和H中格的一些结果，它们加强了[ 11 ]和[ 15]中的一些结果。若H是非紧型对称空间，且D是Io（H）的子群，则D的对偶条件由D的Selberg性质（S）所隐含[20，pp. 4-6]或[10]。在[10]中得到了一个部分匡威。这是一个有趣的问题，这两个条件在这种情况下是否等价。我们的密度结果与[5]的结果非常相似。在命题4.2中，我们得到了波莱尔密度定理的微分几何形式（参见：[5]的推论4.2）：设H不允许Euclidean de Rham因子，G_I（H）是一个子群，其正规化子D在I（H）中满足对偶条件。则（1）G是离散的或（2）存在流形H1，/ /2使得（a）H等距于黎曼积H1 × H2，（B）H1是非紧型对称空间，（c）（~）0=Io（H1）和（d）存在离散子群B~_I（Hz），其在1（//2）中的正规化子满足对偶条件，使得Io（HO·是t子群）在0中具有有限指数。利用上述结果或第3节的主要定理，我们得到了其等距群I（H）满足对偶条件的流形H的如下分解（命题4.1）：设I（H）满足对偶条件。然后存在流形H 0，Ht和H2，其中两个可能具有零维，使得（1）H等距于
Let H denote a complete simply connected Riemannian manifold of nonpositive sectional curvature, and let I(H) denote the group of isometries of H. In this paper we consider density properties of subgroups D~_I(H) that satisfy the duality condition (defined below), These density properties also yield characterizations of Riemannian symmetric spaces of noncompact type and results about lattices in H that strengthen several of the results of [ 11 ] and [ 15]. If H is a symmetric space of noncompact type and if D is a subgroup of Io(H), then the duality condition for D is implied by the Selberg property (S) for D [20, pp. 4-6] or [10]. A partial converse is obtained in [10]. It is an interesting question whether the two conditions are equivalent in this context. Our density results are very similar to those of [5]. In Proposition 4.2 we obtain a differential geometric version of the Borel density theorem (cf. Corollary 4.2 of [5]): Let H admit no Euclidean de Rham factor, and let G~_I(H) be a subgroup whose normalizer D in I(H) satisfies the duality condition. Then either (1) G is discrete or (2) there exist manifolds Hi , / /2 such that (a) H is isometric to the Riemannian product HlXH2, (b) H1 is a symmetric space of noncompact type, (c) ((~)0=Io(Hl) and (d) there exists a discrete subgroup B~_I(Hz), whose normalizer in 1(//2) satisfies the duality condition, such that Io(HO• is a subgroup of t) of finite index in 0 . Using the result just quoted or the main theorem of section 3 we then obtain the following decomposition of a manifold H whose isometry group I(H) satisfies the duality condition (Proposition 4.1): Let I(H) satisfy the duality condition. Then there exist manifolds H0, Ht and H2, two of which may have dimension zero, such that (1) H is isometric to