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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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I am not sure if I understand your definition. A ray $\gamma:[0,\infty)\to M$ starting from $p$ means a unit speed geodesics such that $\gamma(0)=p$. Then what does it mean by "the asymptote $\widetilde{\gamma}$ starting from $x\in M$"? Is it the same as the ray? – Paul Jul 5 '12 at 5:24
    
Actually for question 1, I just want find an example to show that "An asymptotic ray emanating from a nxed point is not unique in general". This statement comes from introduction of the paper of JIN-WHAN YIM --- "Complete Open Manifolds And Horofunctions" – Peter Hu Jul 5 '12 at 5:50
    
So, questions 1 and 2 are just my translation from this paper, Maybe I misunderstand the definition of asymptote. – Peter Hu Jul 5 '12 at 5:54
3  
Then take $M$ to be the Euclidean space $\mathbb{R}^n$ and take any point $x\in\mathbb{R}^n$. Then the ray starting from $x$ is nothing but a straight line starting from $x$. And of course it will not be unique. And the sectional curvature of $\mathbb{R}^n$ is identically zero. – Paul Jul 5 '12 at 5:55
    
Oh! so you solved both question 1 and 2 already! Thank you very much! – Peter Hu Jul 5 '12 at 6:12

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.

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