Math Help - Parallel of surface of revolution

1. Parallel of surface of revolution

Show that the parallel (cycles) of a surface of revolution are curvature lines!

Thank you very much in advance!

2. Re: Parallel of surface of revolution

By symmetry the direction of the generatrix is a critical point of the function of normal curvature, at a fixed point, so it is one of the principle directions.
So the other principle direction is the one orthogonal to it, that is, the direction of the parallel cycles.