We study the behavior of maximal geodesics in a finitely connected complete two-dimensional Riemannian manifold M admitting curvature at infinity. In the case where M is homeomorphic to ℝ2 the Cohn-Vossen theorem states that the total curvature of M, say c(M), is ≤2π. We already studied the case c(M) < 2π in our previous paper. So we study the behavior of geodesics in M with total curvature 2π in this paper. Next we consider the case where M has nonempty boundary. In order to know the behavior of distant geodesics in M with boundary, it is useful to investigate the 'visual image' of the boundary of M. The latter half of this paper will be spent to study the asymptotic behavior of the visual image of a subset of M with located point tending to infinity.
|Number of pages||32|
|Publication status||Published - 2004 Feb|
- Ideal boundary
- The Gauss-Bonnet theorem
- Total curvature