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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.

##### MSC:
 53C20 Global Riemannian geometry, including pinching
##### Keywords:
negative Ricci curvature; group of isometries
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