Suppose $(M,g)$ is a smooth compact orientable Riemannian manifold of dimension $d \geq 3$ with a smooth boundary $\partial M$ and let $\gamma$ be a maximal geodesic in $M$ starting from a point $p \in \partial M$ in direction $v \in S\partial M$ and exiting the manifold at $q \in \partial M$. Can we always find $\tilde{v}$ arbitrarily close to $v$ such that the geodesic starting at point $p$ and direction $\tilde{v}$ is non-self-intersecting?
Thanks,
Edit. I think this is not true as stated, take a sphere and a point at the pole... but surely this must be true generically? Maybe use transversality theorem?
