Positively curved manifolds with symmetry
Positively curved manifolds with symmetry
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DOI：
10.4007/annals.2006.163.607
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发表时间：
2006-03
影响因子：
4.9
通讯作者：
Burkhard Wilking
中科院分区：
文献类型：
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作者：
Burkhard Wilking
有非常少的例子黎曼流形的正截面曲率已知。事实上，在24维以上的空间中，所有已知的例子都是局部秩为1的对称空间.本文对这一现象作了部分解释，证明了正曲的单连通紧致流形（M，g），只要它的等距群Iso（M，g）是大的，就可达秩单对称空间所给出的同伦.更精确地说，我们首先证明，如果dim（Iso（M，g））i ∈ 2 dim（M）。6，则M切同伦等价于一个秩1对称空间或M是齐次的。其次，我们证明了在18（k +1 <$2）维以上，每个M切同伦等价于一个秩为1的对称空间，其中k > 0表示上齐性，k = dim（M/Iso（M，g））.
There are very few examples of Riemannian manifolds with positive sectionalcurvature known. In fact in dimensions above 24 all known examplesare diffeomorphic to locally rank one symmetric spaces. We give a partialexplanation of this phenomenon by showing that a positively curved, simplyconnected, compact manifold (M,g) is up to homotopy given by a rank onesymmetric space, provided that its isometry group Iso(M,g) is large. Moreprecisely we prove first that if dim(Iso(M,g)) iÝ 2 dim(M) . 6, then M is tangentially homotopically equivalent to a rank one symmetric space or M is homogeneous. Secondly, we show that in dimensions above 18(k +1)2 each M is tangentially homotopically equivalent to a rank one symmetric space, where k > 0 denotes the cohomogeneity, k = dim(M/Iso(M,g)).