2 Takashi SfflOYA
and moreover, by Gauss-Bonnet theorem,
k
7 = 1
When ej e dM\ all maximal geodesies close enough to e} are simple; if K^iMj)
In, then close enough to ej there exist no maximal geodesies; if K^MJ) = In,
one can not say whether such geodesies exist or not; if K^Mj) 2n, there exists
maximal geodesies arbitrarily close to ej (see Conclusion 7.3). When ej e Int(M'),
if /Coo(My) = 0, nothing can be said in the absence of further assumptions (see
Remarks and Examples in 2.2 and 2.3); if K*(Mj) 0, there exist maximal geodesies
arbitrarily close to ej (see Corollary to Theorem A in 2.2 and Conclusion 7.3); if
K^iMj) e (0,+»), then close enough to ej maximal geodesies essentially behaved as
those of a cone having vertex angle equal to
K^{MJ)
in a sense made precise in the
Preliminary Remark of 2.3 (see also Conclusion 7.3); if /c»(M7) = +°o, then close
enough to ej all maximal geodesies are simple (see Theorem B in 2.3 and Conclusion
7.3). Since each Mj is either a Riemannian half plane or a Riemannian half cylinder
and since, according to Remark 3.4.3.2, any Riemannian half cylinder can be
isometrically embedded into a Riemannian plane, the previous general statements are
mere consequences of the main results of this paper concerning Riemannian planes
stated in Chapter 2 and proved in Chapters 4, 5 and 6 and of additional results
concerning Riemannian half planes (see 3.8 and 7.2, see also 7.1 for half cylinders).
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