Pseudoconvexity and geodesic connectedness (Q910014)

Let M be a connected smooth manifold with a linear connection. The manifold M is called pseudoconvex iff the geodesic convex hull of each compact set lies in a larger compact set. M is disprisoning iff for each inextendible geodesic \(c: (a,b)\to M\) and \(t_ 0\in (a,b),\) both of the sets \(\{c(t): a<t\leq t_ 0\}\) and \(\{c(t): t_ 0\leq t<b\}\) fail to have compact closure. The authors prove that if M is pseudoconvex, disprisoning and has no conjugate points, then (I) for each pair of points of M there is at least one geodesic segment joining them, and (II) for each point p in M the map \(\exp_ p: T_ pM\to M\) is a diffeomorphism and hence M is diffeomorphic to \(R^ n\).