## 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)

### MSC:

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

### Keywords:

space of metrics; differentiable manifold; curvature bounds
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