# Download Behavior of Distant Maximal Geodesics in Finitely Connected by Takashi Shioya PDF

By Takashi Shioya

This monograph reports the topological shapes of geodesics outdoor a wide compact set in a finitely hooked up, whole, and noncompact floor admitting overall curvature. whilst the floor is homeomorphic to a aircraft, all such geodesics behave like these of a flat cone. particularly, the rotation numbers of the geodesics are managed by means of the full curvature. obtainable to rookies in differential geometry, but in addition of curiosity to experts, this monograph beneficial properties many illustrations that improve realizing of the most principles.

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**Additional info for Behavior of Distant Maximal Geodesics in Finitely Connected Complete 2 Dimensional Riemannian Manifolds**

**Example text**

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?