By Takashi Shioya

This monograph experiences 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 way of the whole curvature. available to rookies in differential geometry, but in addition of curiosity to experts, this monograph beneficial properties many illustrations that improve figuring out of the most rules.

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3. The case of the half cylinder. Proposition [Col]. e. a complete 2dimensional Riemannian manifold with piecewise smooth boundary dN homeomorphic to the half cylinder S1 * [0,+°°)) such that c(N) is defined. Then xu(A0 = -K(N) - c(N) > 0. Behavior of distant maximal geodesies in 2-dimensional manifolds 43 Proof. 2 to the half plane Cl(N-^. 1. Lemma. Let N be a Riemannian half cylinder. Then there exists a point p in dN and a half geodesic starting atp and properly embedded in TV- dN. Proof. Let {qi} be a sequence of points in N tending to infinity.

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. • Behavior of distant maximal geodesies in 2-dimensional manifolds 51 X*) = X<0 Remark.

Therefore, dD = d(M -D) comes equipped with an intrinsic piecewise smooth structure and is nothing more than 3D in which possibly more than countably many points have been blown up each into finitely many points. 4. Definition. The fine boundary (or in short the boundary) 3D of a closed subset D of a 2-dimensional differentiable manifold M such that Bnd(D) splits into a locally finite union of piecewise smoothly properly embedded curves is the boundary 3D' (in the usual sense) of the manifold C1(M -D) = D* with piecewise smooth boundary (compare the present definition with the analogous one given in [BZ]).