In this note, we look at the difference, or rather the absence of a difference, between the space of metrics of positive scalar curvature and metrics of non-negative scalar curvature. The main tool to analyze the former on a spin manifold is the spectral theory of the Dirac operator and refinements thereof. This can be used, for example, to distinguish between path components in the space of positive scalar curvature metrics. Despite the fact that non-negative scalar curvature a priori does not have the same spectral implications as positive scalar curvature, we show that all invariants based on the Dirac operator extend over the bigger space. Under mild conditions we show that the inclusion of the space of metrics of positive scalar curvature into that of non-negative scalar curvature is a weak homotopy equivalence.

在本说明中，我们着眼于正标量曲率指标与非负标量曲率指标之间的差异，或者说没有差异。在自旋流形上分析前者的主要工具是Dirac算子的光谱理论及其改进。例如，这可用于区分正标量曲率度量空间中的路径分量。尽管非负标量曲率先验并不具有与正标量曲率相同的频谱影响，但我们表明，基于Dirac算子的所有不变量都在更大的空间上扩展。在温和的条件下，我们表明，将正标量曲率的度量空间包含到非负标量曲率的空间是弱同伦等效性。