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Metrics of negative Ricci curvature. (English) Zbl 0824.53033
Using some deformation techniques the author is able to construct Riemannian metrics \(g\) of negative Ricci curvature \(r(g)\) and to prove in this way the following remarkable results: (i) For any \(n \geq 3\) there exist constants \(a(n) > b(n) > 0\) such that any manifold \(M\) with \(\dim M \geq 3\) admits a complete Riemannian metric \(g\) for which \(-a(n) < r(g) < -b(n)\). (ii) Any manifold \(M\) with \(\dim M \geq 3\) admits a complete Riemannian metric \(g\) such that \(r(g) < -1\) and \(\text{Vol} (M,g) < \infty\). (iii) If \(M\) is closed, \(\dim M \geq 3\) and \(G \subset \text{Diff} (M)\) is a finite group, then \(G\) coincides with the group of isometries of a Riemannian metric \(g\) on \(M\) with \(r(g) < 0\).
Roughly speaking, the construction starts with a Riemannian metric \(g^ -_ 3\) on \(\mathbb{R}^ 3\) such that (1) \(r(g^ -_ 3) < 0\) on the unit ball \(B_ 1 (0)\), and (2) \(g\) coincides with the standard Euclidean metric outside the ball.
Then, the analogous metrics \(g^ -_ n\) on \(\mathbb{R}^ n\) are obtained by induction for \(n = 4, 5, \dots\) Finally, Ricci negatively curved balls are distributed over the whole \(M\) to get Riemannian metrics with the desired properties.

53C20 Global Riemannian geometry, including pinching
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