DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§3. Some Global Results 33ξΓ⊥ξvα(s)TφvFigure 3.8“multi-parametrization” of Σ that gives us∫∫ L ∫ 2π#(Γ ∩ ξ ⊥ )dξ =∂F∥Σ0 0 ∂s × ∂F∂φ∥ dφds.Compute that∂F∥ ∂s × ∂F∂φ∥ = | sin φ| (this is the hard part) and finish the proof. (Hints: Aspictured in Figure 3.8, show v(s, φ) =cos φT(s)+sin φ(α(s) × T(s)) is the tangent vector tothe great circle ξ ⊥ and deduce that F(s, φ) =α(s) × v(s, φ). Show that ∂F ∂vand α ×∂φ ∂s areboth multiples of v.)12. Complete the details of the proof of the indicated step in the proof of Theorem 3.5, as follows.Pick an interior point s 0 ∈ ∆.a. Choose ˜θ(s 0 )sothat h(s 0 )= ( cos ˜θ(s 0 ), sin ˜θ(s 0 ) ) . Use the procedure of the proof ofLemma 3.6 to determine ˜θ uniquely as a function that is continuous on each ray −→ s 0 s forevery s ∈ ∆.b. Choose s 1 ∈ ∆. Show first (using the fact that a continuous function on the interval s 0 s 1is uniformly continuous) that there is δ>0sothat whenever ‖s − s ′ ‖ 0, show that there is a neighborhood U of s 1 so that whenever s ∈ U we have‖s − s 1 ‖