4

Takashi SfflOYA

semi-regular in a sense given in 1.4 (see Chapter 4). Theorem D (see 2.3) and its

corollaries give additional results in the case where M does not have too much positive

curvature and in the case where M has no negative curvature. Corollary to Theorem A

asserts that a Riemannian plane has maximal geodesies arbitrarily close to infinity, a

result which guarantees that Theorems B, C and D are not empty. Theorem A stated in

2.2 and proved in Chapter 6 asserts that the visual diameter of any compact set K C

M looked at from a point p e M tends to zero when p tends to infinity. It seems

that it is not possible to prove Corollary to Theorem A without a control of visual

diameters. Moreover such a result is extended to that for unbounded K, which

implies that the number K^H) controls whether in a Riemannian half plane H a

maximal geodesic arbitrarily close to infinity exists or not (see 7.2). The result about

the visual images of unbounded K will be published independently.

Since all the proofs given in the present paper are derived from the Gauss-Bonnet

formula and from Cohn-Vossen's theorem, the statements of this paper clearly extend

to G-surfaces in the sense of Busemann (see [Bu]). Although new, our results should

be considered as elementary. For this reason our presentation tried to be as self-

contained as reasonably possible in order to make the article accessible to beginners.

In Chapter 1 we defined semi-regular, almost regular and regular curves in order to

be able in Chapter 2 to state the main results concerning Riemannian planes. In Chapter

3 we introduce a suitable notion of boundary in order to generalize the Gauss-Bonnet

theorem. In the same chapter we also generalize Cohn-Vossen's theorem to a large

class of complete 2-dimensional Riemannian manifolds. Chapter 4 (resp. 5) shows that

outside a fat (resp. fat enough) subset of a Riemannian plane M, all maximal

geodesies are semi-regular (resp. almost regular with suitable index). Chapter 6 proves

Theorem A (the statement concerning visual diameter). Chapter 7 generalizes the

previous results to finitely connected Riemannian manifolds with finitely connected

boundary.