By Takashi Shioya
This monograph reports the topological shapes of geodesics outdoors a wide compact set in a finitely hooked up, entire, and noncompact floor admitting overall curvature. whilst the floor is homeomorphic to a aircraft, all such geodesics behave like these of a flat cone. specifically, the rotation numbers of the geodesics are managed by way of the full curvature. obtainable to novices in differential geometry, but in addition of curiosity to experts, this monograph positive aspects many illustrations that improve knowing of the most principles.
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Extra resources for Behavior of Distant Maximal Geodesics in Finitely Connected Complete 2 Dimensional Riemannian Manifolds
Let 7 [0,+°°) —» M be an improper nonclosed half geodesic in M -K, where M is a strict Riemannian plane and K a contractible compact subset of M. If Q := Cl(7) - y CL M is entirely contained in AT, then Q is just the image of a simple closed geodesic 7such that 7= Bnd(AT) = dK. Proof. As noticed above, Q ^ 0 and 42 n Int(AT) = 0. Moreover Q is a topologically connected compact subset of M whose arcwise connected components are simple maximal or closed geodesies contained in Bnd(7T) such that no path of K is allowed to cross any of these components.
6. Remark. As shown by the example of the Riemannian plane M4 constructed below, the condition C2': "M+ is bounded and M~ is unbounded and c(M) = if' is dubious in the sense that outside any large enough compact subset K of M the geodesies may satisfy Condition (i) or (ii). 2). Example of a dubious Riemannian plane. 3), so that A contains the fourth quadrant of R 2 . The curve c := dA is the image of a smooth proper embedding of R into R 2 containing the negative y axis and asymptotic to the positive x axis with only one inflection point pa = (a,g2(a)) with a e (1,2).
The connected components My of M - M Q are in 1-1 correspondence with the ends e ; ofM, where 1 < j < k. 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.