If $M$ is a convex-cocompact hyperbolic 3-manifold, and $S$ is a closed surface with genus $\geq$ 2. Suppose $f:S\to M$ is a minimal immersion, and $f(S)$ is negatively curved. I know that all the closed geodesics in $f(S)$ are closed geodesics in $M$. Can I conclude that $f(S)$ is totally geodesic in $M$?



Totally geodesic submanifold of a hyperbolic 3-manifold
Reading the first paragraph of the introduction of the following paper:
<http://homeweb.unifr.ch/parlierh/pub/BuserParlierOsaka.pdf>
it seems to me that the set of unit tangent vectors $v\_p$ to $S$ such that the $\gamma(t):=exp(tv\_p)$ is a closed geodesic is dense. If I am not misunderstanding such a density then $\alpha(v\_p,v\_p) = 0$ for a dense set of unit vectors, where $\alpha$ is the second fundamental form of $f(S)$. Thus, $f(S)$ is totally geodesic in $M$ since $\alpha(v\_p,v\_p) \equiv 0 $ for all $v\_p$ due to the density.


