  Dedicated to Professor K. Shiohama on the occasion of his seventieth
birthday: This article is the third in a series of our investigation on a
complete non-compact connected Riemannian manifold $M$. In the first series
[arXiv:0901.4010], we showed that all Busemann functions on an $M$ which is not
less curved than a von Mangoldt surface of revolution are exhaustions, if the
total curvature of the surface is greater than $\pi$. A von Mangoldt surface of
revolution is, by definition, a complete surface of revolution homeomorphic to
Euclidean plane whose Gaussian curvature is non-increasing along each meridian.
Our purpose of this series is to generalize the main theorem in
[arXiv:0901.4010] to an $M$ which is not less curved than a more general
surface of revolution.
