Riemann Curvature Tensor

The Riemann curvature tensor $R$ measures how much parallel transport around a small loop rotates vectors. This is the quantitative measure of the curvature of a manifold.

The Riemann tensor is defined in terms of the Lie bracket and covariant derivative:

$R(u, v)w = \nabla_u \nabla_v w - \nabla_v \nabla_u w - \nabla_{[u,v]} w$

In components, using the Christoffel symbols:

$R^k{}_{l ij} = \partial_i \Gamma^k_{jl} - \partial_j \Gamma^k_{il} + \sum_m\left(\Gamma^k_{im}\Gamma^m_{jl} - \Gamma^k_{jm}\Gamma^m_{il}\right)$

Sphere Example

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

$R^\theta{}_{\phi\theta\phi} = \sin^2\theta$

At $\theta = \pi/2$: $R^\theta{}_{\phi\theta\phi} = 1$.

At $\theta = \pi/4$: $R^\theta{}_{\phi\theta\phi} = \sin^2(\pi/4) = 0.5$.

Gaussian Curvature

For a 2D surface, the Riemann tensor has one independent component. The Gaussian curvature is:

$K = \frac{R^\theta{}_{\phi\theta\phi}}{g_{\phi\phi}} = \frac{\sin^2\theta}{\sin^2\theta} = 1$

For the unit sphere, $K = 1$ everywhere — it is a surface of constant positive curvature.

Flat space has $K = 0$: all $\Gamma = 0$, so all $R = 0$.

Your Task

Implement riemann(g_fn) that returns a function at(point) computing $R[k][l][i][j] = R^k{}_{lij}$.