On the existence of a straight line (Q1107131)

For a complete, noncompact Riemannian manifold M, a geodesic \(\gamma\) satisfying \(d(\gamma (t_ 1),\gamma (t_ 2))=| t_ 1-t_ 2|\) for all \(t_ 1,t_ 2\in R\), where d is a distance function induced from the Riemannian metric on M, is called a straight line on M. In 1936 \textit{S. Cohn-Vossen} [Rec. Math. Mosc., N. Ser. 1, 139-163 (1936; Zbl 0014.27601)] proved that if a complete Riemannian manifold homeomorphic to \(R^ 2)\) admits total curvature \(C(M)=\int_{M}Gd_ M\), and if there exists a straight line on M, then \(C(M)\leq 0\), where G is the Gaussian curvature with respect to the volume element of M. Recently, \textit{K. Shiohama} [J. Differ. Geom. 23, 197-205 (1986; Zbl 0599.53047)] generalized this result and showed that \(C(M)\leq 2\pi (\chi (M)-1)\) if M has only one end, where \(\chi(M)\) is the Euler characteristic of M. The author proves the converse, i.e.: If M is a connected, complete, noncompact, oriented and finitely connected Riemannian 2-manifold having one end and admitting total curvature C(M) which is smaller than \(2\pi (\chi(M)-1),\) then M contains a straight line.