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For an n-dim noncompact manifold M without boundary. Assume M contains a line $\gamma$. For every point $p \in M$, let $\widetilde{p\gamma(t)}$ be the geodesic from p to $\gamma(t)$.

Choose a subsequence $t_i \to \infty$, such that $\widetilde{p\gamma(t_i)}$ converge to a ray $\gamma_p$ from p to infty which is asymptotic to $\gamma$ .

Consider the Buseman function $\beta (\cdot)=\lim_{t\to \infty}(t-|\gamma(t)\cdot|)$. Then along the ray $\gamma_p$, $\beta$ grows with the speed 1. So we get $\beta$ is regular at every point p in M with $|\nabla\beta(p)|=1$ ?

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  • $\begingroup$ what if you have two (different) rays from p asymptotic to \gamma? $\endgroup$
    – valeri
    Apr 6, 2014 at 9:28
  • $\begingroup$ @valeri:Choose any one of them. $\endgroup$
    – wang mu
    Apr 6, 2014 at 9:31
  • $\begingroup$ then your regular function would have two gradients? $\endgroup$
    – valeri
    Apr 6, 2014 at 9:36
  • $\begingroup$ I didn't get the question. Busemann function clearly might have many critical points. But it does has gradient of unit length almost everywhere. $\endgroup$
    – J. GE
    Apr 8, 2014 at 13:53

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