Curvature \(h\)-principles. (English) Zbl 0909.58005

This paper derives some properties of the spaces of all metrics on a differentiable manifold \(M\) which satisfy curvature bounds. In particular, the spaces \(\text{Ric}^{<\alpha}(M)\) and \(S^{<\alpha}(M)\) of metrics on \(M\) with, respectively, Ricci curvature less than \(\alpha\) and scalar curvature less than \(\alpha\) for \(\alpha\in\mathbb{R}\) are investigated. Theorem A says that these spaces are contractible, while Theorem B is an approximation result that these spaces are dense in the space of all metrics in the \(C^0\) and the Hausdorff topologies. Thus, any metric is \(C^0\) close to a metric of arbitrarily negative Ricci curvature. Theorems C and D are local versions of these results.
Reviewer: P.Michor (Wien)


58D17 Manifolds of metrics (especially Riemannian)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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