Asymptotic Curves and Geodesics on Surfaces

In this chapter we begin a study of special curves lying on surfaces in R. An asymptotic curve on a surface M ⊂ R is a curve whose velocity always points in a direction in which the normal curvature of M vanishes. In some sense, M bends less along an asymptotic curve than it does along a general curve. As a simple example, the straight lines on the cylinder (u, v) 7→ (cosu, sinu, v) formed by setting u constant are asymptotic curves. If p is a hyperbolic point of M (meaning that the Gaussian curvature is negative at p), there will be exactly two asymptotic curves passing through p. In Section 18.1, we derive the differential equation that must be satisfied in order that a curve be asymptotic. This is obtained by merely substituting the velocity vector of the curve into the second fundamental form, and setting the result equal to zero. We also prove Theorem 18.7, which relates the Gaussian curvature of M to the torsion (as defined in Section 7.2) of any asymptotic curve lying on M. In Section 18.2 we identify families of asymptotic curves on various classes of surfaces, and construct patches built out of asymptotic curves.