I have come across a question, which I am not convinced I have the right answer to.
Let (M,g) be a connected, non-compact, complete Riemannian manifold, $p\in M$.
a) Show that there exists a sequence $(x_i)_{i\in \mathbb{N}}$ with $d(p,x_i)\overset{i\to \infty}{\longrightarrow} \infty$
b) Show that there exists $X_i\in T_pM$ with $|| X_i ||=1$ so that $x_i = exp_p(d(p,x_i)X_i)$
where $d:M\times M \to \mathbb{R}$ is the distance function.
For a) I thought, that since $M$ is non-compact, there exists a diverging geodesic $\gamma$ with $\gamma(0) = p$. Because $M$ is complete its length is $\infty$. Is that correct reasoning? In class we have not talked about non-compact manifolds.
For $b)$ I thought about using the fact that $exp_p$ is a radial isometry and sends straight lines through $0\in T_pM$ to geodesics through $p\in M$.
Thanks in advance
Best Answer
Unfortunately no, you can't use that for (a) because it is circular reasoning (how do you propose to prove the existence of the ray $\gamma$ otherwise?). But (b) follows from standard equivalent definitions of completeness (e.g. Hopf-Rinow).