Generalized Toponogov comparison theorem for manifolds of roughly non-negative radial curvature

Let M be a complete open Riemannian manifold. A pair (M, o) of M and a point o is an element of E M is said to be dominated by a non-negative continuous function h : [0, infinity) -> [0, infinity) if it satisfies the following. That is, the minimal radial curvature at p is an element of M from o is not less than -h (d (o, p)), where d is the distance function of M. In the case where integral(infinity)(0) r . h(r)dr < infinity, we say that (M, o) or M is of roughly non-negative radial curvature. hi this article, we study a generalized Toponogov's theorem and topology of manifolds of roughly non-negative radial curvature.