  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.
