arbitrary large finite number of homologically invisible closed geodesic the same way as before). Remark that in our examples closed geodesics are without self-intersections and not contractible as implied by the theorem. Counter-examples where the theorem fails by lack of completeness can be found as well by choosing embedded cylinders of revolution restricting the domain of the embedding (\[eq:revolution\]) to $(\theta,z)\in S^1\times (a,b)$ for $a,b\in{{\mathbb{R}}}$. We could proceed as follows: take an even $r:[-1,1]\to(0,+\infty)$ with $r'>0$ on $[-1,0)$ such that $z=0$ is the only closed geodesic of the associated compact embedded cylinder. One can find such an $r$ by slightly modifying a Tannery surface: a sufficient condition is that the metric $g$ on the interior of the cylinder can be written as $$g = \left[ \alpha + h(\cos\rho)\right]^2{\mathrm{d}}\rho^2 
    + \sin^2 \rho\,{\mathrm{d}}\theta^2,$$ for a good choice of coordinates $(\rho,\theta)\in(0,\pi)\times S^1$, where $\alpha$ is irrational and $h:(-1,1)\to(-\alpha,\alpha)$ is a smooth odd function (see for instance [@Bes78 Theorem 4.13]). Then extend $r$ to a smooth map $(-3,1]\to(0,+\infty)$ with $r|_{(-3,-1)} < r(-1)$, $r'<0$ on $(-3,-2)$ and $r'>0$ on $(-2,-1)$. Then $z=-2$ and $z=0$ are the only closed geodesic of the cylinder embedded by $r|_{(-3,1)}$ and are both visible.

In a similar way, we can 