DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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48 Chapter 2. Surfaces: Local Theory(and, similarly, ≥ k 2 ). Moreover, as the Spectral Theorem tells us, the maximum and minimumoccur at right angles to one another. Looking back at Figure 2.2, where the slices are taken at anglesin increments of π/8, we see that the normal slices that are “most curved” appear in the third andseventh frames; the asymptotic directions appear in the second and fourth frames. (Cf. Exercise8.)Next we come to one of the most important concepts in the geometry of surfaces:Definition. The product of the principal curvatures is called the Gaussian curvature: K =det S P = k 1 k 2 . The average of the principal curvatures is called the mean curvature: H = 1 2 trS P =12 (k 1 + k 2 ). We say M is a minimal surface if H =0.Note that whereas the signs of the principal curvatures change if we reverse the direction of theunit normal n, the Gaussian curvature K, being the product of both, is independent of the choiceof unit normal. (And the sign of the mean curvature depends on the choice.)Definition. Fix P ∈ M. We say P is an umbilic 3 if k 1 = k 2 . If k 1 = k 2 =0,wesayP is aplanar point. IfK =0but P is not a planar point, we say P is a parabolic point. IfK>0, we sayP is an elliptic point, and if K
§2. The Gauss Map and the Second Fundamental Form 49rectangle collapses to a line segment; for a saddle surface, orientation is reversed by n and so theGaussian curvature is negative.)Let’s close this section by revisiting our discussion of the curvature of normal slices. Suppose αis an arclength-parametrized curve lying on M with α(0) = P and α ′ (0) = V. Then the calculationin formula (†) onp.44shows thatII P (V, V) =κN · n;i.e., II P (V, V) gives the component of the curvature vector κN of α normal to the surface M atP , which we denote by κ n and call the normal curvature of α at P . What is remarkable aboutthis formula is that it shows that the normal curvature depends only on the direction of α at Pand otherwise not on the curve. (For the case of the normal slice, the normal curvature is, up to asign, all the curvature.) We immediately deduce the followingProposition 2.5 (Meusnier’s Formula). Let α be acurve on M passing through P with unittangent vector V. ThenII P (V, V) =κ n = κ cos φ,where φ is the angle between the principal normal, N, ofα and the surface normal, n, atP .In particular, if α is an asymptotic curve, then its normal curvature is 0 at each point.Example 6. Let’s now investigate a very interesting surface, called the pseudosphere, asshownin Figure 2.6. It is the surface of revolution obtained by rotating the tractrix (see Example 2 ofnNFigure 2.6Chapter 1, Section 1) about the x-axis, and so it is parametrized byx(u, v) =(u − tanh u, sechu cos v, sechu sin v),u > 0, v∈ [0, 2π).Note that the circles (of revolution) are lines of curvature: Either apply Exercise 13 or observe,directly, that the only component of the surface normal that changes as we move around the circle isnormal to the circle in the plane of the circle. Similarly, the various tractrices are lines of curvature:In the plane of one tractrix, the surface normal and the curve normal agree.Now, by Exercise 1.2.5, the curvature of the tractrix is κ =1/ sinh u; since N = n along thiscurve, we have k 1 = κ n =1/ sinh u. Now what about the circles? Here we have κ =1/sechu =cosh u, but this is not the normal curvature. The angle θ we see in Figure 1.9 of Chapter 1 isthe same as the angle φ between N and n in Figure 2.6 (to see why, see Figure 2.7). Thus, by
Page 1: DIFFERENTIALGEOMETRY:A First Course
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Page 8 and 9: 6 Chapter 1. CurvesAlternatively, s
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Page 18 and 19: 16 Chapter 1. CurvesFigure 2.2Conve
Page 20 and 21: 18 Chapter 1. Curvesalso Exercise 2
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Page 30 and 31: 28 Chapter 1. Curvesh(s,t)α(t)α(s
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98 Chapter 3. Surfaces: Further Top
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100 Chapter 3. Surfaces: Further To
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108 Appendix. Review of Linear Alge
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ANSWERS TO SELECTED EXERCISES1.1.1
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116 SELECTED ANSWERS2.4.8 Only circ
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118 INDEXisometry, 107K, 48k-point
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DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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