Orbihedra of nonpositive curvature
Orbihedra of nonpositive curvature
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非正曲率Orbihedra
DOI：
10.1007/bf02698640
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发表时间：
1995
期刊：
影响因子：
--
通讯作者：
M. Brin
中科院分区：
文献类型：
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作者：
W. Ballmann;M. Brin
非正曲率的二维奥比面体是一对 (X, Г)，其中 X 是具有分段平滑度量的二维单纯复形，使得 X 具有 Alexandrov 和 Busemann 意义上的非正曲率，并且 Г 是 X 的一组等距，其行为正确地不连续且协紧。通过与非正曲率的黎曼流形类比，我们引入了 (X, Γ) 的阶 1 的自然概念，结果证明它仅依赖于 Γ 并证明，如果 X 是无边界的，则 (X, Γ) 的阶为 1，或者 X 是两棵树的乘积，或者 X 是一个厚的欧几里得建筑。在第一种情况下，X 上的测地线流是拓扑传递的，并且闭合测地线是稠密的。
A 2-dimensional orbihedron of nonpositive curvature is a pair (X, Γ), where X is a 2-dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and Γ is a group of isometries of X which acts properly discontinuously and cocompactly. By analogy with Riemannian manifolds of nonpositive curvature we introduce a natural notion of rank 1 for (X, Γ) which turns out to depend only on Γ and prove that, if X is boundaryless, then either (X, Γ) has rank 1, or X is the product of two trees, or X is a thick Euclidean building. In the first case the geodesic flow on X is topologically transitive and closed geodesics are dense.