Abstract

In this paper we extend results on the geometry of manifolds with asymptotically nonnegative curvature to manifolds with asymptotically nonnegative minimal radial curvature, showing that most of the results obtained by [U. Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology. Ann. Sci. École Norm. Sup. (4) 18 (1985), 651–670. MR839689 (87j:53058) Zbl 0595.53043], [A. Kasue, A compactification of a manifold with asymptotically nonnegative curvature. Ann. Sci. École Norm. Sup. (4) 21 (1988), 593–622. MR982335 (90d:53049) Zbl 0662.53032] and [S.-H. Zhu, A volume comparison theorem for manifolds with asymptotically nonnegative curvature and its applications. Amer. J. Math. 116 (1994), 669–682. MR1277451 (95c:53049) Zbl 0953.53027] hold in a more general context. Particularly, we show that there exists one and only one tangent cone at infinity to each such manifold, in contrast with the class of manifolds of nonnegative Ricci curvature.