Complete open manifolds of non-negative radial curvature (Q1338923)

Let \(M\) be a complete, connected Riemannian manifold and let \(x\in M\). Suppose that \(\delta\) is a lower bound for the sectional curvature of those planes \(\sigma\) in \(M\) that are tangent to a minimal geodesic from \(x\) to the foot point of \(\sigma\). Then triangle comparison holds for geodesic triangles with a vertex in \(x\). The author concludes, among other things, that the set of critical points of the distance function to \(x\) is bounded if \(\delta=0\).