Generic Properties of Geodesic Flows on Convex Hypersurfaces of Euclidean Space

Andrew Clarke

Published 2019-08-13Version 1

Consider the geodesic flow on a real-analytic, closed, and strictly convex hypersurface $M$ of $\mathbb{R}^n$, equipped with the Euclidean metric. The flow is entirely determined by the manifold and the Riemannian metric. Typically, geodesic flows are perturbed by perturbing the metric. In the present paper, only the Euclidean metric is used, and instead the manifold $M$ is perturbed. In this context, analogues of the following theorems are proved: the bumpy metric theorem; a theorem of Klingenberg and Takens regarding generic properties of $k$-jets of Poincar\'e maps along geodesics; and the Kupka-Smale theorem. Moreover, the proofs presented here are valid in the real-analytic category. Together, these results imply the following two main theorems: there is a $C^{\omega}$-open and dense set of real-analytic, closed, and strictly convex surfaces $M$ in $\mathbb{R}^3$ on which the geodesic flow with respect to the Euclidean metric has a hyperbolic periodic orbit with a transverse homoclinic orbit; and if $M$ is a real-analytic, closed, and strictly convex hypersurface in $\mathbb{R}^n$ (with $n \geq 3$) on which the geodesic flow with respect to the Euclidean metric has a nonhyperbolic periodic orbit, then $C^{\omega}$-generically the geodesic flow on $M$ with respect to the Euclidean metric has a hyperbolic periodic orbit with a transverse homoclinic orbit.