  We investigate the finiteness structure of a complete non-compact
$n$-dimensional Riemannian manifold $M$ whose radial curvature at a base point
of $M$ is bounded from below by that of a non-compact von Mangoldt surface of
revolution with its total curvature greater than $\pi$. We show, as our main
theorem, that all Busemann functions on $M$ are exhaustions, and that there
exists a compact subset of $M$ such that the compact set contains all critical
points for any Busemann function on $M$. As corollaries by the main theorem,
$M$ has finite topological type, and the isometry group of $M$ is compact.
