Let $S \subseteq \mathbb{R}^n$ be a smooth embedded submanifold. Consider $S$ as a Riemannian manifold, with the induced metric which comes from the standard Euclidean metric on $\mathbb{R}^n$.
Is it possible that there exist two geodesics $\gamma,\beta$ in $S$, starting at some $p \in S$, such that $\ddot \gamma(0)=-\ddot \beta(0) \neq 0$?
Here $\ddot \gamma(0),\ddot \beta(0)$ are the extrinsic accelerations, i.e. the accelerations of $\gamma,\beta$ when we view them as paths in $\mathbb{R}^n$.