DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

64 Chapter 2. Surfaces: Local Theoryparallel postulate, which stipulates that given any line L in the plane and any point P not lying onL, there is a unique line through P parallel to L). It seems quite intuitive to say that, working justin R 3 ,two vectors V (thought of as being “tangent at P ”) and W (thought of as being “tangent atQ”) are parallel provided that we obtain W when we move V “parallel to itself” from P to Q; inother words, if W = V. But what would an inhabitant of the sphere say? How should he compareVWPQAre V and W parallel?Figure 4.1a tangent vector at one point of the sphere to a tangent vector at another and determine if they’re“parallel”? (See Figure 4.1.) Perhaps a better question is this: Given a curve α on the surface andavector field X defined along α, should we say X is parallel if it has zero derivative along α?We already know how an inhabitant differentiates a scalar function f : M → R, byconsideringthe directional derivative D V f for any tangent vector V ∈ T P M.Wenow begin with aDefinition. We say a function X: M → R 3 is a vector field on M if(1) X(P ) ∈ T P M for every P ∈ M, and(2) for any parametrization x: U → M, the function X◦x: U → R 3 is (continuously) differentiable.Now, we can differentiate a vector field X on M in the customary fashion: If V ∈ T P M,wechoose a curve α with α(0) = P and α ′ (0) = V and set D V X =(X◦α) ′ (0). (As usual, the chainrule tells us this is well-defined.) But the inhabitant of the surface can only see that portion of thisvector lying in the tangent plane. This brings us to theDefinition. Given a vector field X and V ∈ T P M,wedefine the covariant derivative∇ V X =(D V X) ‖ = the projection of D V X onto T P M = D V X − (D V X · n)n.Given a curve α in M, wesay the vector field X is covariant constant or parallel along α if∇ α ′ (t)X = 0 for all t. (This means that D α ′ (t)X =(X◦α) ′ (t) isamultiple of the normal vectorn(α(t)).)Example 1. Let M be asphere and let α be agreat circle in M. The derivative of the unittangent vector of α points towards the center of the circle, which is in this case the center of thesphere, and thus is completely normal to the sphere. Therefore, the unit tangent vector field ofα is parallel along α. Observe that the constant vector field (0, 0, 1) is parallel along the equatorz =0of a sphere centered at the origin. Is this true of any other constant vector field? ▽