DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

32 Chapter 1. Curvesɛ2ɛ 1ɛ 3CFigure 3.78. Let C be a C 2 closed space curve, say parametrized by arclength by α: [0,L] → R 3 . A unitnormal field X on C is a C 1 vector-valued function with X(0) = X(L) and X(s) · T(s) =0and‖X(s)‖ =1for all s. Wedefine the twist of X to betw(C, X) = 12π∫ L0X ′ (s) · (T(s) × X(s))ds.a. Show that if X and X ∗ are two unit normal fields on C, then tw(C, X) and tw(C, X ∗ )differ by an integer. The fractional part of tw(C, X) (i.e., the twist mod 1) is called thetotal twist of C. (Hint: Write X(s) =cos θ(s)N(s)+sin θ(s)B(s).) ∫1b. Prove that the total twist of C equals the fractional part of τds.2π Cc. Prove that if a closed curve lies on a sphere, then its total twist is 0. (Hint: Choose anobvious candidate for X.)Remark. W. Scherrer proved in 1940 that if the total twist of every closed curve on asurface is 0, then that surface must be a (subset of a) plane or sphere.9. A convex plane curve can be determined by its tangent lines (cos θ)x+(sin θ)y = p(θ), called itssupport lines. The function p(θ) iscalled the support function. (Here θ is the polar coordinate.)a. Prove that the line given above is tangent to the curve at the pointα(θ) =(p(θ) cos θ − p ′ (θ) sin θ, p(θ) sin θ + p ′ (θ) cos θ).b. Prove that the curvature of the curve at α(θ) is1 /( p(θ)+p ′′ (θ) ) .c. Prove that the length of α is given by L =∫ 2π0p(θ)dθ.∫ 2πd. Prove that the area enclosed by α is given by A = 1 (p(θ) 2 − p ′ (θ) 2) dθ.2 0e. Use the answer to part c to reprove the result of Exercise 7.10. (See Exercise 1.2.22.) Under what circumstances does a closed space curve have a parallel curvethat is also closed? (Hint: Exercise 8 should be relevant.)11. Prove Proposition 3.2 as follows. Let α: [0,L] → Σbethe arclength parametrization of Γ,and define F: [0,L] × [0, 2π) → ΣbyF(s, φ) =ξ, where ξ ⊥ is the great circle making angle φwith Γ at α(s). Check that F takes on the value ξ precisely #(Γ ∩ ξ ⊥ ) times, so that F is a