Takashi Shioya's Behavior of Distant Maximal Geodesics in Finitely Connected PDF

By Takashi Shioya

ISBN-10: 082182578X

ISBN-13: 9780821825785

This monograph experiences the topological shapes of geodesics outdoor a wide compact set in a finitely attached, entire, and noncompact floor admitting overall curvature. while 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 additionally of curiosity to experts, this monograph beneficial properties many illustrations that increase knowing of the most rules.

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

Example text

2. Notice that Bp is a smooth revolution ball of axis ( p ) x R with north pole (xo,yo,l/2) and south pole (*o>;yo>-l/2). The boundary dB4 of the set J54 := U Bp is a smooth Riemannian plane M4 C R 3 such that K^MA) = n. The surface M4 is flat on the Behavior of distant maximal geodesies in 2-dimensional manifolds 23 subset (Ax {1/2}) U (A x {-1/2}) U D, where D C M4 is defined by the conditions x < 0 and y < 0. Notice that for all p e c, M4 n Bp is the meridian jUp of Bp whose projection on R 2 is the normal vector to c of length e pointing outside A.

Let G be a nonempty closed subset —which can be split into a finite union of piecewise smoothly properly embedded curves— in a noncompact differentiable manifold M (possibly with boundary) and let A{G) be the union of G and of all bounded connected components of M - G. Notice that Bnd(^i(G)) is a closed subset of G (in general not equal to G) admitting a splitting of the same kind. 4, A(G) *• G), but that if A{G) is connected, G is not necessarily connected (for instance let G be the union of two concentric distinct circles in R 2 ).

In a Riemannian plane M such that (as in Theorem D) the positive total curvature c+{M) is less than In, any maximal geodesic y is semiregular. 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.

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Behavior of Distant Maximal Geodesics in Finitely Connected Complete 2 Dimensional Riemannian Manifolds by Takashi Shioya

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