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Proposition. Let AT be a quasi-fat subset of a strict Riemannian plane M. 1). Proof. In the present proof, we assume that all geodesic arcs o~ are parameterized in such a way that I 6(0 I = 1 for each value t of the parameter. Recall first that a nonclosed geodesic arc y nas t0 be weakly transversely immersed. Let 70 be a closed geodesic in M. 3, c(A(yo)) > 2n, so that 70 has to intersect any quasi-fat subset KQ of M and 70 - (70 n KQ) is the union of transversely immersed open geodesic arcs starting and ending at KQ.

3 is semi-regular) contains K in the interior of its teardrop. 7. Proposition. Let AT be a fat compact subset of a strict Riemannian plane M. Then any geodesic arc of M - K starting at K is simple. Proof. Let y [0,a] —> M be a transversal immersion such that 7(0) e K and that y((0,a)) C M - K has at least one crossing point, let b e (0,a) be the smallest value of the parameter of the curve such that 7 restricted to (0,6] is not simple and let c e (0,6) be such that y{c) = y{b). Now the disk bounded by the loop y([c,b]) C M - K has to be either a teardrop not intersecting AT or a tomato containing K, so that according to (Bl) and (B2), y([b,c]) can not be a geodesic loop.

3 can be applied in order to conclude. 5. Ideal boundary and curvature at infinity. 1) with respect to any side of a defined and finite. Two elements a and p of C are said to be pseudo-parallel when one of the two following disjoint conditions holds. (PP1) A(a u p) is not bounded. (PP2) The fine closure of one of the two unbounded components of M - A(a u P U c) is a half plane H such that K^H) = n, where c is a suitable compact arc intersecting a and p. Notice that this is an equivalence relation on t?

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