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Parallel transport problem

  1. Sep 15, 2010 #1
    I've finally decided to tackle a diff geom book, but I'm having trouble with this Problem 4/Chapter 2 from Do Carmo's Riemannian Geometry:

    Let [tex]M^2\subset R^3[/tex] be a surface in [tex]R^3[/tex] with induced Riemannian metric. Let [tex]c:I\rightarrow M[/tex] be a differentiable curve on [tex]M[/tex] and let [tex]V[/tex] be a vector field tangent to [tex]M[/tex] along [tex]c[/tex]; [tex]V[/tex] can be thought of as a smooth function [tex]V:I\rightarrow R^3[/tex], with [tex]V(t)\in T_{c(t)}M[/tex].

    a)show that [tex]V[/tex] is parallel if and only if [tex]dV/dt[/tex] is perpendicular to [tex]T_{c(t)}\subset R^3[/tex] where [tex]dV/dt[/tex] is the usual derivative of [tex]V:I\rightarrow R^3[/tex]

    b) hopefully I can handle myself. will come back if not! :)

    So I guess the plan is to use
    [tex]DV/dt=(dv^k/dt + \Gamma^k_{ij} v^j dx^i/dt) X_k=0[/tex]
    and dot it with some vector [tex]u^iX_i[/tex]. If I can show that the second term in Dv/dt dotted with this u is zero the problem is done, but I don't see why that should be true..
    if I dot [tex]X_i[/tex] with [tex]X_j[/tex] i get [tex]\delta_{i,j}[/tex], right? what then?

    any hints would be great! Thanks!
  2. jcsd
  3. Sep 16, 2010 #2
    And did you solve Problem 3 before getting to Problem 4?
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