We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed n-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric are examples of closed manifolds with an enlargeable Riemannian length-structure. Moreover, the result is correct in arbitrary dimensions based on the main result of a recent paper by Schoen and Yau. We define the positive MV-scalar curvature on closed orientable topological manifolds and show the compactly enlargeable length-structures are the obstructions of its existence.

我们定义了封闭拓扑流形上的可扩展长度结构，然后表明具有可扩展黎曼长度结构且具有任意闭合光滑流形的闭合n流形的连接总和不包含标量曲率为正的黎曼度量。我们表明，具有局部CAT（0）-metric高度等效于Riemannian度量的闭合光滑歧管是具有扩大的Riemannian长度结构的闭合歧管的示例。而且，基于Schoen和Yau最近的论文的主要结果，该结果在任意维度上都是正确的。我们定义了封闭的可定向拓扑流形上的正MV-标量曲率，并表明紧密可扩展的长度结构是其存在的障碍。