Singular Spaces of Non-Positive Curvature
Singular Spaces of Non-Positive Curvature
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非正曲率奇异空间
DOI：
10.1007/978-1-4684-9167-8_10
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发表时间：
1990
期刊：
影响因子：
--
通讯作者：
W. Ballmann
中科院分区：
文献类型：
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作者：
W. Ballmann
在[Gr 5 j格罗莫夫解释了定义上曲率界限的奇异空间，一个概念可以追溯到AD亚历山德罗夫，比照。[ABNj.下面是对这些材料的讨论。主要应用是判定某些单连通多面体的双曲性，由下面的定理7和14可以得出：单连通局部紧完备测地空间X是双曲的，如果它有曲率I（y：; X<0. lat的含义。t. er通过CAT不等式（C=比较，A= Alexandrov，T= Toponogov）定义，这在第1节中讨论。第二节讨论了非正曲率空间，证明了Cartan-Hadamard定理的一个版本（定理13和14）。在第三节中，我们考虑了k-单形与唯一复形中的单形等距的单纯复形。e是具有常截面曲率X的单连通k维黎曼流形l 'vI ~。定理15最后给出了当这样的空间具有曲率：; X时的一个（充分必要的）判据，它与我们上面所说的结合给出了上面提到的双曲性的判据。
In [Gr5j Gromov explains the definition of upper curvature bounds for singular spaces, a concept which goes back to AD Aleksandrov, cf.[ABNj. Below is a discussion of this material. The main application is a criterion for the hyperbolicity of certain simply connected polyhedra.From Theorems 7 and 14 below it follows that a simply connected locally compact complete geodesic space X is hyperbolic if it has curvature I (y:::; X< O. The meaning of the lat. t. er is defined via the CAT-inequality (C= comparison, A= Alexandrov, T= Toponogov) which is discussed in section 1. In section 2 we disc1l8s spaces of non-positive curvature and prove a version of the Cartan-Hadamard Theorem (Theorems 13 and 14). In section 3 we consider simplicial complexes whose k-simplices, for all k, are isometric to simplices in the unique complet. e simply connected k-dimensional Riemannian manifold l'vI~ of constant sectional curvature X. Theorem 15 finally gives a (necessary and sufficient) criterion when such a space has curvature:::; X which combined with what we said above gives the criterion for hyperbolicity mentioned above.