Let $M$ be a complete* Riemannian manifold. Let $\gamma$ be a geodesic which stops minimizing at some point.
Is it true that $\gamma$ must be periodic or self-intersect? (in a transversal way)
I suspect it's false, but I could not find an example.
*For non complete manifolds it's certainly false (Look at $\mathbb{S}^2 \setminus \{p\}$)
Let $M =\{ (e^{it}, e^{is}) : s, t\in \mathbb R\}$ be the torus with Euclidean metric. Let $$\gamma (t) = (e^{it}, e^{i\sqrt 2 t}).$$ This geodesic never intersect itself. Note also that every geodesics on a compact manifold must stops minimizing at some points.