| n this dissertation we will work with a complete open manifold M of nonnegative curvature. We present a detailed study involving various quantitative aspects of the global geometry of rays and arbitrary geodesics in M, primarily in the case of surfaces. Our results extend and build on the pioneering work of S. Cohn-Vossen as well as the basic ideas in the qualitative structure theory for such spaces given by J. Cheeger, W. T. Meyer, and D. Gromoll.;On a surface, it is a basic problem to understand how an arbitrary geodesic g behaves near infinity. We introduce a concept of asymptotic "winding" and discuss various results in this direction. For example, all geodesics have finite winding for total curvature less than 2;Given r and B, we obtain a family of rays associated with B, called B-rays, which pass through every point of M. We develop the relationship between the B-rays and arbitrary geodesics. Restricting our attention to surfaces, it seems very important to analyze what happens at singularities of B. We introduce the notion of a B-wedge, i.e., a region W bounded by two B-rays which meet at their common initial point. We discuss the total curvature of W. Given that B;Finally, we consider two rays r,r and their associated Busemann functions B,B, respectively, in the case of total curvature equal to 2;Our most important tool is the Busemann function B:M... |
