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Pinching theorem on asymptotically hyperbolic spaces. (English) Zbl 0810.53032
Let $$\overline{M}$$ be a compact manifold with boundary and let $$M$$ be the interior of $$\overline {M}$$. A Riemannian metric $$g$$ defined on $$M$$ is said to be a conformally compact metric if there is a defining function $$\rho$$ for the boundary and a smooth metric $$h$$ on $$\overline {M}$$ such that $$g = {1\over \rho^ 2} h$$. A conformally compact metric $$g$$ is said to be a generalized Poincaré metric if $$g$$ is Einstein and $$h$$ is an even function of a defining function. In this paper we prove the following:
Theorem 1. For any odd integer $$n \geq 2$$, let $$g$$ be a generalized Poincaré metric on $$B^{n + 1} = \{x \in \mathbb{R}^{n + 1} : | x| < 1\}$$. Suppose the sectional curvatures of $$(B^{n + 1},g)$$ approach $$-1$$ in order $$O(\rho^{\tau})$$ for $$\tau > 2$$, then $$(B^{n + 1}, g)$$ is isometric to the standard hyperbolic space.

##### MSC:
 53C20 Global Riemannian geometry, including pinching
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