Stabilities of geodesics on torus (Q2765158)

By using Clairaut formula, the authors describe the global behavior of geodesics on a torus in Euclidean space \(R^3\). They discuss the stability problem of filament winding. From local point of view, the geodesic is most stable on a curved surface. That is, the shape of a flexible thin thread tightened on a given surface is a geodesic, and it does not deform. However, from the global point of view, when a flexible thin thread is tightend, there arise two kinds of unstability. The first comes from the case where several geodesics pass through two given points. The other comes from the case where the filament is laid on the concave side of the torus, in this situation it will produce the ``bridging'' phenomenon. These two kinds of unstability do not fit in filament winding.