Behavior of Distant Maximal Geodesics in Finitely Connected by Takashi Shioya

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.

Show description

Read Online or Download Behavior of Distant Maximal Geodesics in Finitely Connected Complete 2 Dimensional Riemannian Manifolds PDF

Best science & mathematics books

Harmonic Analysis on Totally Disconnected Sets

Booklet by way of Benedetto, John

Analysis und mathematische Physik

Von 1974 bis 1979 hatte ich an der Friedrich-Schiller-Universitat in Jena die sicherlich nicht alltagliche Gelegenheit, einen durchgehenden 10semestrigen Kurs flir Mathematikstudenten zu lesen. Entsprechend dem Studienplan hatten diese Vorlesungen verschiedene N amen (Differential- und Integralrechnung, gewohn liche Differentialgleichungen usw.

50 Visions of Mathematics

Chill out: not anyone is aware technical arithmetic with no long education yet all of us have an intuitive take hold of of the tips at the back of the symbols. To have a good time the fiftieth anniversary of the founding of the Institute of arithmetic and its purposes (IMA), this e-book is designed to exhibit the great thing about arithmetic - together with photos encouraged by way of mathematical difficulties - including its unreasonable effectiveness and applicability, with out frying your mind.

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

Sample text

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]).

Download PDF sample

Rated 5.00 of 5 – based on 18 votes
 

Author: admin