Parallel Transport

When you move a vector along a curve on a manifold while keeping it "as parallel as possible", you perform parallel transport. This is the covariant version of "keeping a vector constant".

Given a curve $\gamma(t)$ with velocity $\dot{\gamma}^i$, the parallel transport equation for a vector $V^k$ is:

$\frac{dV^k}{dt} + \Gamma^k_{ij}\, \dot{\gamma}^i\, V^j = 0$

Flat space: all $\Gamma = 0$, so $dV/dt = 0$ — vectors don't change.

Sphere Example: Transport Along a Meridian

Consider the unit sphere with $\theta$ as colatitude and $\phi$ as longitude. Transport the vector $V = [0, 1]$ (pointing east) along the meridian $\phi = 0$, decreasing $\theta$ from $\pi/2$ toward $0$.

The curve: $\gamma(t) = (\pi/2 - t,\; 0)$, so $\dot{\theta} = -1$, $\dot{\phi} = 0$.

The equations become: $\frac{dV^\theta}{dt} = 0$ $\frac{dV^\phi}{dt} = \Gamma^\phi_{\theta\phi}\, V^\phi = \cot(\theta)\, V^\phi$

Starting from $V = [0, 1]$ at $\theta = \pi/2$, after time $t$:

$V^\phi(t) = \frac{1}{\cos t}$

At $t = \pi/3$: $V^\phi = 1/\cos(\pi/3) = 2.0$. The coordinate component grows, but the physical length stays 1: $|V|^2 = \sin^2(\pi/6) \cdot 4 = 1$.

Your Task

Implement parallel_transport(gamma_dot, V0, christoffel_fn, t_end, steps) using RK4 integration. It should return the final transported vector.