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  • $\begingroup$ Right, I could not make it $C^\infty$, but it seems I can do it $C^2$... $\endgroup$ Commented Jul 15, 2011 at 14:00
  • $\begingroup$ Anton, I believe indeed now, that there can be a counterexample in the class $C^2$, and indeed, if it exists, there will be examples with curves of arbitrary small curvature. $\endgroup$ Commented Jul 15, 2011 at 16:00