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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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