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2, we see that certain normal slices of a saddle surface are true lines. This leads us to make the following Definition. 3 A curve in M is called an asymptotic curve if its tangent vector at each point is an asymptotic direction. Example 3. If a surface M contains a line, that line is an asymptotic curve. For the normal slice in the direction of the line contains the line (and perhaps other things far away), which, of course, has zero curvature. 4. There is an asymptotic direction at P if and only if k1 k2 ≤ 0.

Challenge) What does it mean for L to have 4-point contact with M at P ? 3. The Codazzi and Gauss Equations and the Fundamental Theorem of Surface Theory We now wish to proceed towards a deeper understanding of Gaussian curvature. We have to this point considered only the normal components of the second derivatives xuu , xuv , and xvv . Now u , Γv , let’s consider them in toto. Since {xu , xv , n} gives a basis for R3 , there are functions Γuu uu u = Γ u , Γ v = Γ v , Γ u , and Γ v so that Γuv vu uv vu vv vv u v xuu = Γuu xu + Γuu xv + ℓn (†) u v xuv = Γuv xu + Γuv xv + mn u v xvv = Γvv xu + Γvv xv + nn.

Parametrized Surfaces and the First Fundamental Form 41 if angles measured in the uv-plane agree with corresponding angles in TP M for all P . We leave it to the reader to check in Exercise 5 that this is equivalent to the conditions E = G, F = 0. Since T | | | | E F xu · xu xu · xv = = xu xv xu xv , F G xv · xu xv · xv | | | | we have xu · xu xu · xv 0 xu · xu xu · xv EG − F 2 = det = det xv · xu xv · xv 0 xv · xu xv · xv 0 0 1 T 2 | | | | | | | | | = det xu xv n xu xv n = det xu xv n , | | | | | | | | | which is the square of the volume of the parallelepiped spanned by xu , xv , and n.