DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§2. The Gauss Map and the Second Fundamental Form 47On a sphere, all normal slices have the same (nonzero) curvature. On the other hand, if we lookcarefully at Figure 2.2, we see that certain normal slices of a saddle surface are true lines. Thisleads us to make the followingDefinition. If the normal slice in direction V has zero curvature, i.e., if II P (V, V) =0, thenwe call V an asymptotic direction. 2 A curve in M is called an asymptotic curve if its tangent vectorat each point is an asymptotic direction.Example 3. If a surface M contains a line, that line is an asymptotic curve. For the normalslice in the direction of the line contains the line (and perhaps other things far away), which, ofcourse, has zero curvature. ▽Corollary 2.4. There is an asymptotic direction at P if and only if k 1 k 2 ≤ 0.Proof. k 2 =0if and only if e 2 is an asymptotic direction. Now suppose k 2 ≠0. IfV isa unit asymptotic vector making angle θ with e 1 , then we have k 1 cos 2 θ + k 2 sin 2 θ =0,and sotan 2 θ = −k 1 /k 2 ,sok 1 k 2 ≤ 0. Conversely, if k 1 k 2 ≤ 0, take θ with tan θ = ± √ −k 1 /k 2 , and thenV is an asymptotic direction. □Example 4. We consider the helicoid, as pictured in Figure 1.2. It is a ruled surface and sothe rulings are asymptotic curves. What is quite less obvious is that the family of helices on thesurface are also asymptotic curves. But, as we see in Figure 2.4, the normal slice tangent to thePFigure 2.4helix at P has an inflection point at P , and therefore the helix is an asymptotic curve. We ask thereader to check this by calculation in Exercise 5. ▽It is also an immediate consequence of Proposition 2.3 that the principal curvatures are themaximum and minimum (signed) curvatures of the various normal slices. Assume k 2 ≤ k 1 . Thenk 1 cos 2 θ + k 2 sin 2 θ = k 1 (1 − sin 2 θ)+k 2 sin 2 θ = k 1 +(k 2 − k 1 ) sin 2 θ ≤ k 12 Of course, V ̸=0 here. See Exercise 21 for an explanation of this terminology.