Exponential map of the punctured plane
= Exponential map of the punctured plane
For $S=(\mathbb R^2\setminus\{0\})\times\{0\}\subset\mathbb R^3$ and $p=(1,0,0)$, geodesics are straight lines until they hit the missing origin. Thus $(-2,0,0)\notin\mathcal D_p$, and $(-1,0,0)$ is not in the image of $\exp_p$ because its unique straight segment from $p$ passes through the origin.