Focal points and the decrease of curvature: A surprising example (Q699268)

Let \(K\) be the Gaussian curvature of a surface \(S\) and \(\gamma\) a closed unit speed geodesic on \(S\). Suppose that there are no focal points along \(\gamma\) and that \(K\) is strictly decreasing as we move away from \(\gamma\) along geodesics perpendicular to \(\gamma\) in a small neighborhood of \(\gamma\). For the Fermi coordinates \((s,x)\) along \(\gamma(s)\), we assume (1) \(( \partial K/ \partial x)\) \((s,0)\equiv 0\), (2) \((\partial^2K/ \partial x^2)\) \((s, 0)<0\) for all \(s\). The author shows, by an intuitive reasoning, that for \(T\) sufficiently large, there are no focal points along the restriction of \(\sigma\) to \((-\infty, -T)\), where \(\sigma\) is another unit speed geodesic such that \(\lim_{t \to-\infty} \text{dist} (\sigma(t)\), \(\gamma(t))=0\). The author proves that contrary to his above reasoning, there is an example of a surface \(S\) containing a closed geodesic \(\gamma\) along which there are no focal point and that conditions (1) and (2) are satisfied but there are focal points along every geodesic \(\sigma\) that is asymptotic to \(\gamma\) as \(t\to-\infty\).