Generic properties of geodesic flows on analytic hypersurfaces of Euclidean space

Consider the geodesic flow on a real-analytic closed hypersurface \begin{document}$ M $\end{document} of id="M2">\begin{document}$ \mathbb{R}^n $\end{document}, equipped with induced metric. How commonly can we expect such flows to have transverse homoclinic orbit? In this paper, give following two partial answers question: style='text-indent:20px;'>● If id="M3">\begin{document}$ is in id="M4">\begin{document}$ (with id="M5">\begin{document}$ n \geq 3 $\end{document}) which respect metric has nonhyperbolic periodic orbit, then id="M6">\begin{document}$ C^{\omega} $\end{document}-generically id="M7">\begin{document}$ hyperbolic orbit orbit; and There id="M8">\begin{document}$ $\end{document}-open dense set real-analytic, closed, strictly convex surfaces id="M9">\begin{document}$ id="M10">\begin{document}$ \mathbb{R}^3 orbit. style='text-indent:20px;'>These are among first perturbation-theoretic results for flows.