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.
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