I would like to know if it is possible to have a simple curve $\gamma $ on a surface $S$ such that
- $\gamma$ is compact and embedded (i.e. with respect to the topology induced from $S$ it is homeomorphic to the closed interval $[0,1]$, or to $S^1$)
- $\gamma$ is not locally flat (i.e. $(S,\gamma)$ is not locally homeomorphic to $(\mathbb{R}^2,\mathbb{R})$ or to $(\mathbb{R}^2,[0,+\infty[)$ as topological pair )
Notice that here $\gamma$ is not assumed to be a smooth submanifold; we only require it to be homeomorphic to $[0,1]$ or $S^1$ as a topological manifold (even if $S$ is equipped with a smooth structure). Since the easiest examples of non locally flat embeddings I know come from knots or knotted arcs in 3-dimensional space, I suspect the answer could be negative, but I am not really sure how to prove it. My guess is that having dimension 1 and codimension 1 should prevent wild behaviour.
Thanks in advance for any reference/suggestion/example.