Let $M$ be a smooth closed Riemannian manifold, and let $p \in M$. Given $v \in T_pM$, let $\gamma_{p,v}$ be the geodesic emanating from $p$ with initial velocity $v$.
Define $A=\{ v \in T_pM | \, \gamma_{p,v} \, \text{ is periodic } \}$. Is $A$ always dense in $T_pM$?
I am especially interested in dimension $2$. The example I have in mind is the flat torus $\mathbb{T}=\mathbb{S}^1 \times \mathbb{S}^1$ where the velocities with rational values are dense.