开流形非正曲率的障碍
Obstructions to nonpositive curvature for open manifolds
摘要：
研究了群G上的代数条件，在此条件下，Hadamard流形上的每一个真不连续的等距G-作用都有G-不变的Busemann函数。对于这样的G，我们证明了以下结构定理：同伦等价于余维>2的有限复形的开完备非正曲黎曼K（G，1）流形是相同余维的子复形的开正则邻域。在这个设置中，我们表明，每个切同伦类型包含无限多个开放的K（G，1）流形，不承认完全非正弯曲的度量，即使他们的普遍覆盖是欧几里得空间。一个例子应用是开可缩流形W同胚于欧几里得空间当且仅当W与圆的乘积容许非正曲率的完备黎曼度量。
英文摘要：
We study algebraic conditions on a group G under which every properly discontinuous, isometric G-action on a Hadamard manifold has a G-invariant Busemann function. For such G we prove the following structure theorem: every open complete nonpositively curved Riemannian K(G,1) manifold that is homotopy equivalent to a finite complex of codimension >2 is an open regular neighborhood of a subcomplex of the same codimension. In this setting we show that each tangential homotopy type contains infinitely many open K(G,1) manifolds that admit no complete nonpositively curved metric even though their universal cover is the Euclidean space. A sample application is that an open contractible manifold W is homeomorphic to a Euclidean space if and only if the product of W and a circle admits a complete Riemannian metric of nonpositive curvature.
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