关于Grove和Searle的一个定理
On a theorem of Grove and Searle
摘要：
Grove和Searle的一个定理直接建立了具有8维或更小有效圆对称群的正曲率2d流形M具有正的欧拉示性数X（M）：不动点集N由偶数维正曲率流形组成并且具有欧拉示性数X（N）=X（M）。它不是空的Berger。如果N有余维2分量，Grove-Searle迫使M在{RP ^2d，S ^2d，CP ^d}中。根据Frankel，不可能有两个余维2的情况。在其余的情况下，Gauss-Bonnet-Chern力都具有正的欧拉示性数。这个简单的证明并没有达到记录10或更少，它使用的方法威尔金，但它促使分析结构的不动点组件N，特别是看看正曲率流形承认一个U（1）或SU（2）对称连接或几乎连接的不动点集N。它们具有惊人的测地线性质：不动点流形N与其每个点的焦散一致，并且测地流是可积的。在完全一般性中，Lefschetz不动点性质X（N）=X（M）和Frankel维数定理dim（M）对于N的两个不同连通分量A、B小于dim（A）+ dim（B）在从较小分量建立M中产生已经很重的约束。有可能S^2d、RP^2d、CP^d、HP^d、OP^2、W^6、E^6、W^12、W^24实际上是偶数维正曲率流形的一个完整列表，这些流形具有连续统对称性。除了射影空间之外，已知情形的欧拉示性数总是1、2或6，其中从2到6的跳跃发生在具有四个不动点分量N=S^2 + S^2 + S^0的Wallach或Eenburg流形W^6，E^6上，这是唯一已知的不属于Grove-Searle形式N=N_1或N=N_1 + p}且N_1连通的情形。
英文摘要：
A theorem of Grove and Searle directly establishes that positive curvature 2d manifolds M with effective circular symmetry group of dimension 8 or less have positive Euler characteristic X(M): the fixed point set N consists of even dimensional positive curvature manifolds and has the Euler characteristic X(N)=X(M). It is not empty by Berger. If N has a co-dimension 2 component, Grove-Searle forces M to be in { RP^2d,S^2d,CP^d }. By Frankel, there can be not two codimension 2 cases. In the remaining cases, Gauss-Bonnet-Chern forces all to have positive Euler characteristic. This simple proof does not quite reach the record 10 or less which uses methods of Wilking but it motivates to analyze the structure of fixed point components N and in particular to look at positive curvature manifolds which admit a U(1) or SU(2) symmetry with connected or almost connected fixed point set N. They have amazing geodesic properties: the fixed point manifold N agrees with the caustic of each of its points and the geodesic flow is integrable. In full generality, the Lefschetz fixed point property X(N)=X(M) and Frankel's dimension theorem dim(M) is less than dim(A) + dim(B) for two different connectivity components A,B of N produce already heavy constraints in building up M from smaller components. It is possible that S^2d, RP^2d, CP^d, HP^d, OP^2, W^6,E^6,W^12,W^24 are actually a complete list of even-dimensional positive curvature manifolds admitting a continuum symmetry. Aside from the projective spaces, the Euler characteristic of the known cases is always 1,2 or 6, where the jump from 2 to 6 happened with the Wallach or Eschenburg manifolds W^6,E^6 which have four fixed point components N=S^2 + S^2 + S^0, the only known case which are not of the Grove-Searle form N=N_1 or N=N_1 + p} with connected N_1.
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