DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§4. Covariant Differentiation, Parallel Translation, and Geodesics 65Example 2. A fundamental example requires that we revisit the Christoffel symbols. Given aparametrized surface x: U → M, wehave∇ xu x u =(x uu ) ‖ =Γ u uux u +Γ v uux v∇ xv x u =(x uv ) ‖ =Γ u uvx u +Γ v uvx v = ∇ xu x v ,∇ xv x v =(x vv ) ‖ =Γ uvvx u +Γ vvvx v .▽andThe first result we prove is the followingProposition 4.1. Given a curve α: [0, 1] → M with α(0) = P and X 0 ∈ T P M, there is aunique parallel vector field X defined along α with X(P )=X 0 .Proof. Assuming α lies in a parametrized portion x: U → M, set α(t) =x(u(t),v(t)) andwrite X(α(t)) = a(t)x u (u(t),v(t)) + b(t)x v (u(t),v(t)). Then α ′ (t) =u ′ (t)x u + v ′ (t)x v (where thethe cumbersome argument (u(t),v(t)) is understood). So, by the product rule and chain rule, wehave∇ α ′ (t)X = ( (X◦α) ′ (t) ) (‖ d (= a(t)xu (u(t),v(t)) + b(t)x v (u(t),v(t)) )) ‖dt( d ‖ ( ) d ‖= a ′ (t)x u + b ′ (t)x v + a(t) u(u(t),v(t)))dt x + b(t)dt x v(u(t),v(t))= a ′ (t)x u + b ′ (t)x v + a(t) ( u ′ (t)x uu + v ′ ) ‖ ((t)x uv + b(t) u ′ (t)x vu + v ′ ) ‖(t)x vv= a ′ (t)x u + b ′ (t)x v + a(t) ( u ′ (t)(Γuux u u +Γuux v v )+v ′ (t)(Γuvx u u +Γuvx v v ) )+ b(t) ( u ′ (t)(Γvux u u +Γvux v v )+v ′ (t)(Γvvx u u +Γvvx v v ) )= ( a ′ (t)+a(t)(Γuuu u ′ (t)+Γuvv u ′ (t)) + b(t)(Γvuu u ′ (t)+Γvvv u ′ (t)) ) x u+ ( b ′ (t)+a(t)(Γuuu v ′ (t)+Γuvv v ′ (t)) + b(t)(Γvuu v ′ (t)+Γvvv v ′ (t)) ) x v .Thus, to say X is parallel along the curve α is to say that a(t) and b(t) are solutions of the linearsystem of first order differential equationsa ′ (t)+a(t)(Γuuu u ′ (t)+Γuvv u ′ (t)) + b(t)(Γvuu u ′ (t)+Γvvv u ′ (t)) = 0(♣)b ′ (t)+a(t)(Γuuu v ′ (t)+Γuvv v ′ (t)) + b(t)(Γvuu v ′ (t)+Γvvv v ′ (t)) =0.By Theorem 3.2 of the Appendix, this system has a unique solution on [0, 1] once we specify a(0)and b(0), and hence we obtain a unique parallel vector field X with X(P )=X 0 . □Definition. If Q = α(1), we refer to X(Q) astheparallel translate of X 0 along α, orthe resultof parallel translation along α.Example 3. Fix a latitude circle u = u 0 (u 0 ≠0,π)onthe unit sphere and let’s calculate theeffect of parallel-translating the vector X 0 = x v starting at the point P given by u = u 0 , v =0,oncearound the circle, counterclockwise. We parametrize the curve by u(t) =u 0 , v(t) =t, 0≤ t ≤ 2π.Using our computation of the Christoffel symbols of the sphere in Example 1 or 2 of Section 3, weobtain from (♣) the differential equationsa ′ (t) =sin u 0 cos u 0 b(t), a(0) = 0