 Solution

For the given wrapped hyperbolic cosine curve,so $\boxed{C\subset S}$. The curve makes one full turn around the circular cylinder, descending from $(-1,0,\cosh \pi )$ to $(1,0,1)$ and ascending to its initial point. Its two parameter endpoints coincide; their tangent vectors $(0,-1,-\sinh \pi )$ and $(0,-1,\sinh \pi )$ do not agree, so the resulting closed curve has a corner there.