论文标题
在格罗夫和塞尔定理上
On a theorem of Grove and Searle
论文作者
论文摘要
格罗夫和塞尔的一个定理直接确定，呈正曲率2D歧管M具有有效的圆形尺寸为8或更小的圆形对称群具有正极特性x（m）：固定点集n n组成均匀的正弯曲歧管，并具有Euler特性x（n）= x（m）。它不是伯杰（Berger）空的。如果n具有共含量2组件，则grove-searle seral力在{rp^2d，s^2d，cp^d}中。弗兰克尔（Frankel）不可能有两个Codimension 2案例。在其余的情况下，高斯 - 骨网切恩迫使所有人都具有正极的特征。 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.在完全的一般性中，对于两个不同的连接组件A，N，N lefschetz固定点属性x（n）= x（m）和弗兰克尔的尺寸dim（m）小于dim（a） + dim（b），对于两个不同的连接组件a，n的n，b的b在从较小的组件中构建m时已经产生了很重的约束。 s^2d，rp^2d，cp^d，hp^d，op^2，w^6，e^6，w^12，w^12，w^24实际上是均匀的均值正曲率歧管的完整列表，即接受连续的对称性。除了投影空间外，已知情况的欧拉特征总是1,2或6，其中2到6的跳跃发生在Wallach或Eschenburg w^6，E^6中，具有四个固定点组件n = s^2 + s^2 + s^0，唯一已知的情况是Grove-searle-searle-searle-searle-searle n = n_1或n = n_1或n = n_1或n_1或p n = n _1 + p。
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.
