Local rigidity of symmetric spaces
Local rigidity of symmetric spaces
复制标题
对称空间的局部刚度
DOI：
10.2307/2001755
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发表时间：
1990
影响因子：
1.3
通讯作者：
W. Ziller
中科院分区：
文献类型：
--
作者：
V. Schroeder;W. Ziller
我们证明了在非紧型或紧型对称空间上的度量是局部刚性的，即如果局部改变度量但保持曲率界，则新度量与旧度量是等距的。我们还证明了秩> 3的对称空间的解析延拓性质。本文的目的是证明一个对称空间是刚性的度量的局部修改，保持适当的曲率界限。我们的结果是由下面的定理Gromov [BGS，？[5]：定理1（Gromov）.设X是秩> 2的单连通非正截面曲率对称空间，Q是X中紧集的补。如果f：Q-Y是全测地等距嵌入，其中Y是具有非正截面曲率的完备单连通黎曼流形，则存在f到全测地等距嵌入Tf：X-Y的唯一扩张。我们将给出这个定理的一个稍微不同的证明，这将使我们能够证明下面的对偶结果：定理2。设X是秩> 2且截面曲率为0 0的单连通对称空间，且dim Y = dim X，若Q是X中半径为7r/30的球的补，则任意等距嵌入f：Q-Y唯一地扩张到等距7：X-Y.如果在上述两个定理中，我们选择Y = X和f = id，我们得到对称空间的期望局部刚性。定理2特别适用于S2 × S，并表明霍普夫猜想至少是局部正确的。定理2中球的最佳半径是凸球的半径（在这些假设下> 7r/2）。事实上，在非正曲率的情况下，Gromov [BGS，？5]1987年11月4日和1988年9月12日编辑收到的修订版。1980年数学学科分类（1985年修订）。小学53C35;中学53C20。第一作者由"Schweizerischer Nationalfond"资助。第二作者部分得到了美国国家科学基金会的资助，并感谢IHES和马克斯普朗克研究所的热情款待。(?) 1990年美国数学学会0002 - 9947/90 $1.00 +$.25每页
We show that on a symmetric space of noncompact or compact type the metric is locally rigid in the sense that if one changes the metric locally but preserves the curvature bounds, then the new metric is isometric to the old one. We also prove an analytic continuation property for symmetric spaces of rank > 3 . The aim of this paper is to show that a symmetric space is rigid under a local modification of the metric which preserves suitable curvature bounds. Our results were motivated by the following theorem of Gromov [BGS, ?5]: Theorem 1 (Gromov). Let X be a simply connected symmetric space of rank > 2 and with nonpositive sectional curvature and Q the complement of a compact set in X. If f: Q -Y is a totally geodesic isometric embedding, where Y is a complete simply connected Riemannian manifold with nonpositive sectional curvature, then there exists a unique extension of f to a totally geodesic isometric embedding Tf: X-Y. We will give a slightly different proof of this theorem, which will then enable us to also prove the following dual result: Theorem 2. Let X be a simply connected symmetric space of rank > 2 and with sectional curvature 0 0, and dim Y = dim X, and if Q is the complement of a ball of radius 7r/30 in X, then any isometric embedding f: Q -Y extends uniquely to an isometry 7: X -Y. If in the above two theorems, we choose Y = X and f = id, we obtain the desired local rigidity of symmetric spaces. Theorem 2 applies in particular to S2 x S and shows that the Hopf conjecture is at least locally correct. The optimal radius for the ball in Theorem 2 would be the radius of a convex ball (> 7r/2 under those assumptions). Indeed, in the case of nonpositive curvature, Gromov [BGS, ?5] proved the following local version of Theorem 1: Received by the editors November 4, 1987 and, in revised form, September 12, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C35; Secondary 53C20. The first author is supported by the "Schweizerischer Nationalfond". The second author is partially supported by a grant from the National Science Foundation and would like to thank the IHES and the Max Planck Institut for its hospitality. (?) 1990 American Mathematical Society 0002-9947/90 $1.00 + $.25 per page