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.