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Question 1

Let $M$ be a complete, noncompact Riemannian manifold, a ray $\gamma:[0,\infty) \rightarrow M$ starting from $p$, and a point $x \in M$ such that the asymptote $\widetilde{\gamma}$ starting from $x$. Does $\widetilde{\gamma}$ must be unique? If not, who can give me a counter example?

Question 2

If the sectional curvature of $M$ satisfies $K_M \leqslant 0$, are asymptotes to a given ray (starting from a given basepoint) unique?

Actually for question 1, I just want find an example to show that "An asymptotic ray emanating from a fixed point is not unique in general". This statement comes from introduction of the paper of JIN-WHAN YIM --- "Complete Open Manifolds And Horofunctions" http://www.mathnet.or.kr/mathnet/kms_tex/370.pdf

Thank you very much!

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    Oh! so you solved both question 1 and 2 already! Thank you very much!2012-07-05

1 Answers 1

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Def : Two normal geodesic rays $c_i$ is asymptotic if $ d(c_1(t),c_2(t))\leq k $ for some $k$ and all $t$

(1) $f(r)$ is strictly increasing function s.t. $ f(0)=0,\ \lim_{r\rightarrow 1}f(r)=\infty $

Consider a surface $ z=f(x^2+y^2),\ x^2+y^2 <1 $ for some $f$ Further assume that it has a positive curvature. Then $ (t,0,f(t^2)),\ 0< c\leq t<\infty $ is a ray. Then at $(0,0,0)$ we have two asymptotes.

(2) Yes it is unique. Given a ray $\gamma$ at $p$, assume that we have two asymptotes at $x$ $\widetilde{\gamma}_i$ Then we have a convex function $d^2 ( \widetilde{\gamma}_1 (s),\widetilde{\gamma}_2(s))$ So it goes to infinity. Hence one of them is not asymptote.