0 or K
k > 0.
In this chapter we investigate the global structure of geometric surfaces, that is, 2-dimensional Riemannian manifolds. We want to know what the possible surfaces are and what they are like. In maximum generality this goal is unrealistic, but under reasonable hypotheses, good results can be obtained.
The central theme of this chapter is the influence of Gaussian curvature on geodesics. A significant preliminary result is that in any surface, the geometry of a neighborhood of a point is completely described by curvature and the geodesics radiating out from that point.
This local result can be extended to show that geodesics grip an entire surface. The first step is to show that geodesics starting at any point p of a connected surface eventually reach every point of that surface (Sec.2). Gaussian curvature controls the spreading and contracting of these geodesics as they radiate out to cover the surface, but their global pattern can be quite complicated.
Nevertheless, by using topological methods
(Sec.4),
considerable global information can be derived from this
pattern.
In particular, we give detailed results in two broad cases:
surfaces with constant curvature, and surfaces whose curvature obeys
either K 0 or K k > 0.