By Takashi Shioya

This monograph stories the topological shapes of geodesics outdoors a wide compact set in a finitely attached, entire, and noncompact floor admitting overall curvature. whilst the floor is homeomorphic to a airplane, all such geodesics behave like these of a flat cone. specifically, the rotation numbers of the geodesics are managed by means of the entire curvature. available to rookies in differential geometry, but in addition of curiosity to experts, this monograph gains many illustrations that increase knowing of the most rules.

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Moreover let y\ and 72 be two nonsimple geodesies of M. Then the interiors of their teardrops have nonempty intersection. Proof. 3 applies to all geodesies in M. LetDi and£>2 De the teardrops of two nonsimple geodesies y\ and 72 in M. 1 c(D[) > n (for i = 1,2) and that by assumption c(D\ U D! n D2) > 0. • Remark. In [Ba], V. Bangert proved that in a Riemannian plane admitting no closed geodesies, all maximal geodesies are proper. 3 such a Riemannian plane M contains no closed geodesies.

4. The general case. Proof of Cohn-Vossen's theorem. Let MQ C M be a core of M, namely a 44 Takashi SfflOYA compact submanifold with piecewise smooth boundary on which M retracts and such that 3MQ contains all the compact components of dM and intersects all the noncompact ones. 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.

4, any geodesic 7 of a strict Riemannian plane M such that c+(M) < 2n is semi-regular. 1, ind(7) < n />rN . 3. 4. Lemma. Suppose there exists a constant c' and a semi-regular geodesic 7 for which k(j) = j for some j > 1 and for which the associated domains BL satisfy c{B-) > c* for 1 < /

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