We study the behavior of maximal geodesies outside a sufficiently large compact set
in a finitely connected, complete and noncompact 2-dimensional Riemannian manifold
(possibly with boundary). The total curvature of such a manifold was first investigated
by S. Cohn-Vossen [Col, Co2]. Assume for simplicity that a complete manifold M is
He proved in [Col] that if the total curvature of M exists, it is
less or equal to 2n. One of our main results (see Theorems B and C in 2.3) states that
if the total curvature of M exists and is strictly less than In, then any maximal
geodesic outside a sufficiently large compact set in M forms almost the shape as that
of a maximal geodesic in a flat cone, and its rotation number (originally due to H.
Whitney [Wh]) is controlled by the total curvature.
Key Words and Phrases: geodesies, the Gauss-Bonnet formula, total curvature.