Positive Curvature, Macroscopic Dimension, Spectral Gaps and Higher Signatures

Positive Curvature, Macroscopic Dimension, Spectral Gaps and Higher Signatures
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DOI：
10.1007/978-1-4612-4098-3_1
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发表时间：
1996
期刊：
--
影响因子：
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通讯作者：
M. Gromov;M. Gromov
M. Gromov;M. Gromov
中科院分区：
其他
文献类型：
--
作者：
M. Gromov;M. Gromov

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结论每一个不同于环面的紧致齐性空间都存在一个不变度量，且Sc> O。(This对于那些非紧的齐性黎曼空间也是如此，其中隐含等距群允许非平凡的紧半单因子。）(f)每个非紧型对称空间V都有Sc：：; 0，且Sc= 0意味着V是黎曼平面（即局部等距于lRn）。(g)每个具有左不变度量的连通非Abel可解李群G有Sc< O。（阿贝尔群是黎曼平坦群，且Sc= 0。
Conclusion. Every compact homogeneous space different from a torus admits an invariant metric with Sc> O.(This is also true for those noncompact homogeneous Riemannian spaces where the implied isometry group admits a nontrivial compact semisimple factor.)(f) Every symmetric space V of non-compact type has Sc::; 0 and Sc= 0 implies that V is Riemannian fiat (ie locally isometric to lRn).(g) Every connected non-Abelian solvable Lie group G with a left invariant metric has Sc< O.(Abelian groups are Riemannian fiat and have Sc= 0.)