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

Lohkamp, Joachim

Ann. Math. (2) 142, No. 3, 457-498 (1995).

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