Question in do Carmo's book Riemannian geometry section 7

Question

I have a question. Please help me.

Assume that $M$ is complete and noncompact, and let $p$ belong to $M$. Show that $M$ contains a ray starting from $p$.

$M$ is a riemannian manifold. It is geodesically and Cauchy sequences complete too. A ray is a geodesic curve that its domain is $[0,\infty)$ and it minimizes the distance between start point to each other points of curve.

Answer 1

Otherwise suppose every geodesic emitting from p will fail to be a segment after some distance s. Since the unit sphere in the tangent plane that parameterizing these geodesics is compact, s has a maximum $s_{max}$. This means that the farthest distance from p is $s_{max}$, among all points of the manifold. So the diameter of the manifold is bounded by $2s_{max}$, by the triangle inequality. So the manifold is bounded and complete, by the Hopf–Rinow theorem, it is then compact.

Answer 2

Take a sequence of points $(x_i)$ in the manifold whose distance from $p$ tends to infinity, and connect each of them to $p$ by a minimizing geodesic $\gamma_i(s)$. Choose a convergent subsequence $\gamma'_{i_k}(0)$ at $p$. Then the limit of the sequence is the desired direction of a ray.