I've been trying to go through an example for parallel transport but I cannot quite follow the solution.
A surface (paraboloid) is given by the parametric equation $r(ρ, φ)$ = $ρ \cos(φ)\hat{i}$ + $ρ \sin(φ)\hat{j}$ + $aρ^2\hat{k}$ where a is a constant. Evaluate the metric and Christoffel symbols in the (ρ, φ) coordinates. A unit vector X initially pointing along $e_φ$ is parallel transported along the curve given by ρ = R, φ = 2πt for 0 ≤ t ≤ 1. What is the angle between the initial X(t = 0) and final X(t = 1) vectors?
I've found the metric: $g_{ρρ} = 1+4a^2ρ^2$, $g_{φφ} = ρ^2$, $g_{φρ} = 0$ and the Christoffel symbols $\Gamma^ρ_{ρρ}=\frac{4a^2ρ}{1+4a^2ρ^2}$, $\Gamma^ρ_{φφ}=\frac{-ρ}{1+4a^2ρ^2}$, $\Gamma^φ_{φρ}=\Gamma^φ_{ρφ}=\frac{1}{ρ}$.
Now using the parallel transport equation, $\dot{X^a} + \Gamma^a_{bc}X^b\dot{x^c} = 0$, I get
$\dot{X^ρ} - \frac{R}{1+4a^2R^2}X^φ\dot{φ}+ \frac{4a^2R}{1+4a^2R^2}X^ρ\dot{ρ} = 0$ and $\dot{X^φ} + \frac{1}{R}X^ρ\dot{φ} + \frac{1}{R}X^φ\dot{ρ}= 0$
Then I am not sure how to proceed - any help would be much appreciated!
Insert the velocity of the curve, $\dot{\rho}=0, \dot{φ}=2\pi$, so that the parallel transport equations become $$\dot{X}^ρ - \frac{2\pi R}{1+4a^2R^2}X^φ= 0\,,\ \ \ \dot{X}^φ + \frac{2\pi}{R}X^ρ= 0.$$ Then solve this system with initial datum $\dot{X}^ρ(0)=0, \dot{X}^φ(0)=1$ taking time derivative of both equations. When you have found the final vector $X(1)$ compute the angle $$\frac{g(X(0),X(1))}{g(X(0),X(0))g(X(1),X(1))}.$$