If $M$ is a graph of a $C^1$ function $f:\mathbb{R}^n\to\mathbb{R}$, is it true that the length minimizing geodesics on $M$ are $C^1$? I expect a counterexample.

For a related discussion see Metric angles in Riemannian manifolds of low regularity and Existence and uniqueness of geodesics in low regularity.