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Negative bending of open manifolds. (English) Zbl 0840.53023
The author revamps his work on the $$h$$-principle for the existence of metrics of negative Ricci curvature $$\text{Ric} < 0$$ (see also the preceding review) to present a new “strong” deformation yielding what he calls the first step in understanding negative Ricci curvature. The results are the following. First, every open manifold admits a conformal deformation to a complete metric with $$\text{Ric} < 0$$. Second, every closed manifold of dimension $$\geq 3$$ admits a metric with $$\text{Ric} < 0$$. The starting point in dimension $$\geq 4$$ is the remark that the Euclidean space contains a closed manifold of codimension 2 with trivial normal bundle and admitting a metric with $$\text{Ric} < 0$$. Third, the local version of the closed case states that the standard metric of $$\mathbb{R}^n$$, $$n \geq 3$$ can be modified inside a unit ball to a metric with $$\text{Ric} < 0$$.

MSC:
 53C20 Global Riemannian geometry, including pinching
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