Non-negative versus positive scalar curvature

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. (C) 2020 The Authors. Published by Elsevier Masson SAS.