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Lower curvature bounds, Toponogov’s theorem, and bounded topology. (English) Zbl 0595.53043
This is the first of two papers devoted to studying the topology of asymptotically non-negatively curved manifolds. Here a complete Riemannian manifold M with base point \(p_ 0\) is said to be asymptotically non-negatively curved, if there is a monotone decreasing function \(\lambda\) : [0,\(\infty)\to [0,\infty)\) such that : \((i)\quad \int^{\infty}_{0}r \lambda (r)dr<\infty\) and (ii) the sectional curvatures at p are bounded below by \(-\lambda (d(p_ 0,p))\). The point of departure is an extension of Toponogov’s triangle comparison theorem to the case where the comparison manifold is not a simply connected surface of constant curvature but a surface with only rotational symmetries. Aside from this the main result in the present paper gives an a priori upper bound for the number of ends of an asymptotically non- negatively curved manifold. The sequel gives an improvement and extension to asymptotically non-negatively curved manifolds, of M. Gromov’s ”Betti number theorem” [Comment. Math. Helv. 56, 179-195 (1981; Zbl 0467.53021)].
Reviewer: K.Grove

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