DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§3. Surface Theory with Differential Forms 99The 1-form ω 12 is called the connection form and measures the tangential twist of e 1 . Just aswe saw in Section 1, then, ∇ V e 1 is the tangential component of D V e 1 = de 1 (V) =ω 12 (V)e 2 +ω 13 (V)e 3 , which is, of course, ω 12 (V)e 2 .From the Gauss equation and Stokes’s Theorem, the Gauss-Bonnet formula follows immediatelyfor an oriented surface M with (piecewise smooth) boundary ∂M on which we can globally definea moving frame. That is, we can reprove the Local Gauss-Bonnet formula, Theorem 1.5, quiteeffortlessly.Proof. We start with an arbitrary moving frame e 1 , e 2 , e 3 and take a Darboux frame e 1 , e 2 , e 3along ∂M. Wewrite e 1 = cos θe 1 +sin θe 2 and e 2 = − sin θe 1 +cos θe 2 (where θ is smoothly chosenalong the smooth pieces of ∂M and the exterior angle ɛ j at P j gives the “jump” of θ as we crossP j ). Then, by Stokes’s Theorem and Lemma 3.3, we have∫∫∫∫∫∫(KdA = − dω 12 = − ω 12 = − ω12 − dθ ) ∫= − κ g ds +(2π − ∑ ɛ j ).M(See Exercise 2.)□M∂M∂M∂MEXERCISES 3.31. Prove Lemma 3.3.2. Let e 1 , e 2 , e 3 be the Darboux frame along a curve α. Show that as a 1-form on α, ω 12 = κ g ω 1 .3. Suppose α is a curve lying in the surface M. Let e 1 , e 2 , e 3 be the Darboux frame along α (i.e.,amoving frame for the surface with e 1 tangent to α), and let e 1 = e 1 , e 2 , e 3 be the Frenetframe. Then, by analogy with Lemma 3.3, e 2 , e 3 are obtained from e 2 , e 3 by rotating throughsome angle θ. Show that, as 1-forms on α, wehave:ω 12 = κω 1 = cos θω 12 + sin θω 13ω 13 = 0 = − sin θω 12 + cos θω 13ω 23 = τω 1 = ω 23 + dθ.*4. Use Exercise 3 to prove Meusnier’s Theorem (Proposition 2.5 of Chapter 2).5. Use Exercise 3 to prove that if C ⊂ M is a line of curvature and the osculating plane of Cmakes a constant angle with the tangent plane of M, then C is planar.6. Using moving frames to redo Exercise 2.2.13.*7. Use moving frames to compute the Gaussian curvature of the torus, parametrized as in Example1(c) of Chapter 2.8. The vectors e 1 = v(1, 0) and e 2 = v(0, 1) give a moving frame at (u, v) ∈ H. Set ω 1 = du/vand ω 2 = dv/v.a. Check that for any V ∈ T (u,v) H, ω 1 (V) =I(V, e 1 ) and ω 2 (V) =I(V, e 2 ).