Behavior of distant maximal geodesies in 2-dimensional manifolds 3

In the case where M is a Riemannian plane (i.e. homeomorphic to

R2)

with the

previously introduced notations ffoo(M) -2n-c(M) e [0,+«] Theorems B and

C stated in Chapters 2 and 3 assert that, if K^M) 0, outside a sufficiently large

contractible compact subset K of M (called 'fat enough' in Chapter 5), all maximal

geodesies in M are regular in a sense given in Chapter 1, which means that they

behave as those of a cone, of a two-sheeted hyperboloid or of a paraboloid. In general,

the number of double points of such geodesies (which have no triple points) is equal to

intg(;r/K"oo(M)) (where intg(-) means the integral part). However, when n I

K"oo(M) is an integer, it may happen that this number drops down to % I fCoo(M) - 1

('the less crossing situation'). Moreover in 'the more crossing situation', i.e. when the

number of double points is equal to n I K^M), it may happen that the geodesic

crosses itself over and over again in such a way that it can no longer be called 'regular'

but only 'almost regular' (see 1.10).

a reguler geodesic an almost regular geodesic

It is interesting in itself (and also useful for the proof) to notice that, if we partly relax

the constrains on the quantities both of positive and of negative curvature that a

contractible compact subset must contain in order to be called fat enough, one obtains a

larger class of subsets K of M called 'fat' outside which all maximal geodesies are