If two embedded surfaces are tangent along a curve, their tangent planes agree there. The curve's ambient acceleration is normal to one surface exactly when it is normal to the other, so it is a geodesic of one exactly when it is a geodesic of the other.
A circular helix on a circular cylinder has radial ambient acceleration and is therefore a geodesic. Any surface tangent to the cylinder along that helix has the same curve as a geodesic.
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