Let $(M, g)$ be a complete Riemannian manifold and let $\gamma : [0, \infty) \to \mathbb{R}$ be a ray, i.e. a unit speed geodesic such that for every $s, t \ge 0$ : $$ dist\big(\gamma(s), \gamma(t)\big) = |s - t|. $$ Let us fix a point $p \in M$. We can define a family $(\sigma_t)_{t \ge 0}$ of segments joining $p$ to $\gamma(t)$. Is it easy to prove that there exist a sequence $(t_i)_i$ such that the segment $(\sigma_{t_i})_i$ converge pointwise to a new ray $\gamma_p$ emanating from $p$. Such a ray is called asymptote for $\gamma$. I know that in general an asymptote is not unique but I would like to see a concrete example.
Thank you!
Consider a $$\{ (x,y,z)| x^2+y^2=1,\ z\geq 0\} $$
We have a complete manifold $M_1$ by making a suitable $cap$ around $z=0$ And around $(x-2+\epsilon )^2+y^2=1$ we do same thing so that we have $M_2$ Here they are intersect along $x= \frac{2-\epsilon}{2}$ So by cutting along the intersection and deleting we attach them And by small perturbation we have smooth manifold $M$ Here for $z<0$ it is like two mountains which are adjacent.
Let $c$ be a ray $(-1,0,t),\ t\geq 0 $ And fix a point $p=(\frac{ 2-\epsilon}{2},0,z_0)$ where $z_0$ is smallest. Then at $p$ we have two asymptotes wrt $c$