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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.

53C20 Global Riemannian geometry, including pinching
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