From do Carmo's Riemannian Geometry P151:

Let M be a complete Riemannian manifold, and let $X$ be a differentiable vector field on $M$. Suppose that there exists a constant $с > 0$ such that $|X(p)| > c$, for all $p \in M$. Prove that the trajectories of $X$, that is, the curves $\phi(t)$ in M with $\phi'(t) = X(\phi(t))$, are defined for all values of $t$.

I think this is wrong. The counterexample is $M=\mathbb{R}$, $X = x^2+1$. Then $\phi(t) = \tan t$ which is not defined everywhere. I think the proof that do Carmo wants is to take $t_n \rightarrow t$, then note that $\phi(t_n)$ approaches a limit point $r$ on $M$, and then to extend the trajectory at $r$. But the issue is that $r$ may not exist since $\phi(t_n)$ may not approach anything, so this method is faulty.

The corrected problem I am proposing is that $0 < |X(p)| < c$. Then the limit point exists, and the proof method goes forward.

I emailed my professor about this, but she says the problem is correct as stated, so it is do Carmo and my professor on one side and just me (an undergraduate) on the other. She told me to "Read Lemma 3.3 on p.150", and that is pretty similar to the proof method I outlined above; it is showing that geodesics extend on complete manifolds. But geodesics have unit speed so $|X(p)|$ is not an issue there.

So my question is: is the problem actually correct as stated?

  • 5
    $\begingroup$ You're right, do Carmo's wrong. Your counterexample is fine. $\endgroup$ – Amitai Yuval Dec 1 '14 at 20:51

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