DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§4. Covariant Differentiation, Parallel Translation, and Geodesics 67Our experience in Example 3 suggests the followingProposition 4.2. Parallel translation preserves lengths and angles. That is, if X and Y areparallel vector fields along a curve α from P to Q, then ‖X(P )‖ = ‖X(Q)‖ and the angle betweenX(P ) and Y(P ) equals the angle between X(Q) and Y(Q).Proof. Consider f(t) =X(α(t)) · Y(α(t)). Thenf ′ (t) =(X◦α) ′ (t) · (Y◦α)(t)+(X◦α)(t) · (Y◦α) ′ (t)= D α ′ (t)X · Y + X · D α ′ (t)Y (1)= ∇ α ′ (t)X · Y + X ·∇ α ′ (t)Y (2)=0.Note that equality (1) holds because X and Y are tangent to M and hence their dot product withany vector normal to the surface is 0. Equality (2) holds because X and Y are assumed parallel alongα. Itfollows that the dot product X·Y remains constant along α. Taking Y = X, weinfer that ‖X‖(and similarly ‖Y‖) isconstant. Knowing that, using the famous formula cos θ = X · Y/‖X‖‖Y‖for the angle θ between X and Y, weinfer that the angle remains constant. □Now we change gears somewhat. We saw in Exercise 1.1.8 that the shortest path joining twopoints in R 3 is a line segment. One characterization of the line segment is that it never changesdirection, so that its unit tangent vector is parallel (so no distance is wasted by turning). It seemsplausible that the mythical inhabitant of our surface M might try to travel from one point toanother in M, staying in M, bysimilarly not turning; that is, so that his unit tangent vectorfield is parallel along his path. Physically, this means that if he travels at constant speed, anyacceleration should be normal to the surface. This leads us to the followingDefinition. We say a parametrized curve α in a surface M is a geodesic if its tangent vectoris parallel along the curve, i.e., if ∇ α ′α ′ =0.Recall that since parallel translation preserves lengths, α must have constant speed, although itmay not be arclength-parametrized. In general, we refer to an unparametrized curve as a geodesicif its arclength-reparametrization is in fact a geodesic.In general, given any arclength-parametrized curve α lying on M, wedefined its normal curvatureat the end of Section 2. Instead of using the Frenet frame, it is natural to consider the Darbouxframe for α, which takes into account the fact that α lies on the surface M. (Both are illustratedin Figure 4.3.) We take the right-handed orthonormal basis {T, n × T, n}; note that the first twoBn×TnNTThe Frenet and Darboux framesFigure 4.3vectors give a basis for T P M.Wecan decompose the curvature vectorκN = ( ) ( )κN · (n × T) (n × T)+ κN} {{ }} {{ · n } n.κ gκ n