The study of the curvature and topology of Riemannian manifolds is mainstream in differential geometry. Many of the important contributions in this topic go back to the pioneering works by Cohn-Vossen in 1935–6, [19] and [20]. In fact the study of total curvature on complete noncompact Riemannian manifolds made by him contains many fruitful ideas. Many hints in his thoughts lead us to the study of the curvature and topology of Riemannian manifolds.

The well-known Gauss–Bonnet theorem states that the total curvature of a compact Riemannian 2-manifold is a topological invariant. Cohn-Vossen first proved that the total curvature of a finitely connected complete noncompact Riemannian 2-manifold M is bounded above by 2πχ(M), where χ(M) is the Euler characteristic of M. Among many beautiful consequences of this result, he proved the splitting theorem for complete open Riemannian 2-manifolds of nonnegative Gaussian curvature admitting a straight line. The structure theorem for such 2-manifolds was also established by him. He investigated the global behavior of complete geodesics on these 2-manifolds and this gave rise to the study of poles. The Bonnesen-type isoperimetric problem for complete open surfaces admitting a total curvature was first investigated by Fiala [26] for the analytic case and then by Hartman [34] for the C2 case. Here the Cohn-Vossen theorem plays an essential role. The total curvature of infinitely connected complete open surfaces was discussed by Huber from the point of view of complex analysis.

The definition of the cut locus was introduced by Poincaré ([68]), and he first investigated the structure of the cut locus of a point on a complete, simply connected and real analytic Riemannian 2-manifold. Myers ([60, 61]) determined the structure of the cut locus of a point in a 2-sphere and Whitehead [107] proved that the cut locus of a point on a complete two-dimensional Riemannian manifold carries the structure of a local tree. In this chapter we will determine the structure of the cut locus and distance circles of a Jordan curve in a complete Riemannian 2-manifold and will prove the absolute continuity of the distance function of the cut locus.

Preliminaries

Throughout this chapter (M, g) always denotes a complete connected smooth two-dimensional Riemannian manifold without boundary. Let γ : [0, ∞) → M be a unit-speed geodesic. Note that any geodesic segment γ : [0, a) → M is extensible to [0, ∞) according to Theorem 1.7.3. If, for some positive number b, γ |[0,b] is not a minimizing geodesic joining its endpoints, let t0 be the largest positive number t such that γ |[0,t] is minimizing. Note that there always exists such a positive number t0, by Lemma 1.2.2. The point γ (t0) is called a cut point of γ (0) along the geodesic γ. For each point p on M, let C(p) denote the set of all cut points along the geodesics emanating from p.

We observed in Chapter 5 that the existence of a total curvature imposes some strong restrictions on the structure of distance circles. In this chapter, we shall see that the total curvature of a finitely connected complete open two-dimensional Riemannian manifold imposes strong restrictions on the mass of rays emanating from an arbitrary fixed point. The first result on the relation between the total curvature and the mass of rays was proved by Maeda in [51]. In [76], Shiga extended this result to the case where the sign of the Gaussian curvature changes. Some relations between the mass of rays and the total curvature were investigated, in detail, by Oguchi, Shiohama, Shioya and Tanaka [62, 83, 84, 90]. Also, Shioya investigated the relation between the mass of rays and the ideal boundary of higher-dimensional spaces with nonnegative curvature (cf. [90]).

Preliminaries; the mass of rays emanating from a fixed point

Let M be a connected, finitely connected, smooth complete Riemannian 2-manifold.

Note that if M contains no straight line (see Definition 2.2.1) then it has exactly one end.

Lemma 6.1.1.Assume that M contains no straight line. Then, for each compact subset K of M, there exists a number R(K) such that if q ∈ M satisfies d(q, K) > R(K) then no ray emanating from q passes through any point on K.

We shall introduce the classical results obtained by Cohn-Vossen in [19] and [20] and Huber in [39] by exhibiting a simpler method under more general assumptions. For this purpose we need to consider the curvature measure over an unbounded domain, not necessarily oriented, with possibly noncompact boundary. The boundary curves may be divergent and hence certain conditions for them will be required. Our ideas will lead to the natural compactification of complete open surfaces for which the Gauss–Bonnet theorem is valid. Some examples on the total curvature of complete open surfaces in R3 are provided. From these examples, one can begin to see the geometric significance of the total curvature of complete open surfaces.

The total curvature of complete open surfaces

From now on let M be a connected, complete and noncompact Riemannian 2-manifold either with or without a piecewise-smooth boundary. In general we cannot expect the Gauss–Bonnet theorem to hold for such an M. However, it is still of interest to consider the Gauss–Bonnet theorem for noncompact surfaces. For this purpose we need to ascertain:

1. how to define the Euler characteristic of M;

2. how to make sense of the curvature integral over M.

For point (1) we are led to consider the notion of the finite connectivity of M.

It is not easy to find a nontrivial pole even on a surface of revolution, unless the latter has a nonpositive Gaussian curvature. We shall give a necessary and sufficient condition for a surface of revolution to have nontrivial poles. The proof is achieved by obtaining Jacobi fields along any geodesic (see [101]). The method is found in a classical work of von Mangoldt [59]. We will also determine the cut loci of a certain class of surfaces of revolution containing well-known examples: the two-sheeted hyperboloids of revolution and the paraboloids of revolution (see [102]). von Mangoldt proved in [59] that any point on a two-sheeted hyperboloid of revolution is a pole if the point is sufficiently close to the vertex. Furthermore, he proved in [59] that the two umbilic points of a two-sheeted hyperboloid are poles and that the poles of any elliptic paraboloid are the two umbilic points. These surfaces are typical examples of a Liouville surface. A definition of global Liouville surfaces was introduced by Kiyohara in [44]. See also [40] for poles on noncompact complete Liouville surfaces.

Properties of geodesics

A surface of revolution means a complete Riemannian manifold (M, g) homeomorphic to R2 that admits a point p such that the Gaussian curvature of M is constant on S(p, t) for each positive t. The point p is called the vertex of the surface of revolution.

Throughout this chapter (M, g) denotes a surface of revolution and p denotes the vertex of the surface.