Parallel transport problem

1. Sep 15, 2010

drgigi

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

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

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

So I guess the plan is to use
$$DV/dt=(dv^k/dt + \Gamma^k_{ij} v^j dx^i/dt) X_k=0$$
and dot it with some vector $$u^iX_i$$. 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 $$X_i$$ with $$X_j$$ i get $$\delta_{i,j}$$, right? what then?

any hints would be great! Thanks!

2. Sep 16, 2010