# Generic Properties of Geodesic Flows on Analytic Hypersurfaces of Euclidean Space

## TL;DR

This paper proves generic properties of geodesic flows on real-analytic hypersurfaces in Euclidean space, showing that typical perturbations lead to hyperbolic periodic orbits with transverse homoclinic orbits, extending classical dynamical results to the real-analytic setting.

## Contribution

It establishes real-analytic versions of key theorems in dynamical systems for geodesic flows on hypersurfaces, including the bumpy metric, Klingenberg-Takens, and Kupka-Smale theorems.

## Key findings

- Generic perturbations produce hyperbolic periodic orbits with transverse homoclinic orbits.
- Results apply to real-analytic hypersurfaces in Euclidean space for dimensions n ≥ 3.
- Methods extend to perturbations of metrics on Riemannian manifolds.

## Abstract

Consider the geodesic flow on a real-analytic closed hypersurface $M$ of $\mathbb{R}^n$, equipped with the standard Euclidean metric. The flow is entirely determined by the manifold and the Riemannian metric. Typically, geodesic flows are perturbed by varying the metric. In the present paper, however, 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 topology. Together, these results imply the following two main theorems: if $M$ is a real-analytic closed 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; and 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. The methods used here also apply to the classical setting of perturbations of metrics on a Riemannian manifold to obtain real-analytic versions of these theorems in that case. These are among the first perturbation-theoretic results for real-analytic geodesic flows.