give examples of complete planes with nothing but an arbitrary finite number of homologically invisible closed geodesics by considering surfaces of revolution (\[eq:revolution\]) parametrized by ${{\mathbb{R}}}/2\pi{{\mathbb{Z}}}\times [0,+\infty)$ with $r:[0,+\infty)\to [0,+\infty)$ being increasing and smooth on $(0,+\infty)$ with $r(0)=0$ and $r'(z)\to +\infty$ when $z\to 0$ in a suitable way (*i.e.* so that the surface is smooth at the origin). Then, as above, we get homologically invisible closed geodesics on the inflexion points of $r$, and nowhere else.

We say that $C_-\subset M$ (resp. $C_+$) is a neighborhood of $-\infty$ (resp. of $+\infty$) if $C_-$ contains $S^1\times (-\infty,a)$ for some $a\in{{\mathbb{R}}}$ (resp. $S^1\times (b,+\infty)$ for some $b\in{{\mathbb{R}}}$) for an arbitrarily fixed identification of $M$ with $S^1\times{{\mathbb{R}}}$. In order to prove Theorem \[thm:main\], we will extensively use the following theorem due to Bangert:

\[thm:Bangert\] Let $M$ be a complete Riemannian cylinder with isolated closed geodesics and suppose there exists disjoint locally convex open neighborhoods $C_-$ and $C_+$ of $-\infty$ and $+\infty$ respectively such that the boundaries $\partial C_\pm$ are not totally geodesic. Then $M$ contains infinitely many homologically visible closed geodesics intersecting $M\setminus (C_-\cup C_+)$ and at least one without self-intersections.

Since Bangert did not give the precise proof of that statement, for 