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!