This monograph studies the topological shapes of geodesics outside a large compact set in a finitely connected, complete, and noncompact surface admitting total curvature. When the surface is homeomorphic to a plane, all such geodesics behave like those of a flat cone. In particular, the rotation numbers of the geodesics are controlled by the total curvature. Accessible to beginners in differential geometry, but also of interest to specialists, this monograph features many illustrations that enhance understanding of the main ideas.

Readership

Graduate students studying differential geometry for the first time as well as specialists in the field.

Table of Contents

• Introduction
• The semi-regular curves in a differentiable plane
• Statement of main results and examples
• Some applications of the Gauss-Bonnet theorem
• Semi-regularity of distant geodesics
• Almost regularity of distant geodesics
• The visual diameter
• Distant geodesics in a finitely connected manifold with finitely connected boundary
• References