Christoffel Symbols

To differentiate a vector field on a manifold, we need a connection — a rule for comparing vectors at different points. The most natural choice for a Riemannian manifold is the Levi-Civita connection, encoded by the Christoffel symbols $\Gamma^k_{ij}$.

From the metric $g_{ij}$, the Christoffel symbols are:

$\Gamma^k_{ij} = \frac{1}{2} g^{kl}\left(\frac{\partial g_{lj}}{\partial x^i} + \frac{\partial g_{li}}{\partial x^j} - \frac{\partial g_{ij}}{\partial x^l}\right)$

where $g^{kl}$ is the inverse metric.

Sphere Example

For the unit sphere with metric $g = \text{diag}(1, \sin^2\theta)$:

$\Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta$ $\Gamma^\phi_{\theta\phi} = \Gamma^\phi_{\phi\theta} = \cot\theta$

All others vanish. At $\theta = \pi/4$:

• $\Gamma^\theta_{\phi\phi} = -\sin(\pi/4)\cos(\pi/4) = -0.5$
• $\Gamma^\phi_{\theta\phi} = \cot(\pi/4) = 1.0$

Flat Space

On flat $\mathbb{R}^n$ with $g_{ij} = \delta_{ij}$, all Christoffel symbols vanish: $\Gamma^k_{ij} = 0$.

Your Task

Implement christoffel(g_fn) that takes a metric function and returns a function at(point) that computes the $2 \times 2 \times 2$ array Gamma[k][i][j] = $\Gamma^k_{ij}$ at that point.