The Metric Tensor

A metric tensor $g$ gives the manifold its geometry — it defines lengths, angles, and volumes. At each point $p$, $g_p$ is a symmetric bilinear form on tangent vectors:

$g(u, v) = \sum_{i,j} g_{ij}(p)\, u^i v^j$

where $g_{ij}$ is the metric component matrix.

Examples

Euclidean metric on $\mathbb{R}^2$: $g_{ij} = \delta_{ij}$ (identity matrix) $g(u, v) = u_x v_x + u_y v_y$

Sphere of radius $R$ in colatitude $\theta$, longitude $\phi$: $g = \begin{pmatrix} R^2 & 0 \\ 0 & R^2\sin^2\theta \end{pmatrix}$

The $\phi$-direction shrinks near the poles ($\theta \to 0$ or $\pi$) because circles of latitude get smaller.

Length of a Vector

$|v|^2 = g(v, v) = \sum_{i,j} g_{ij} v^i v^j$

For $v = [3, 4]$ on flat $\mathbb{R}^2$: $|v|^2 = 9 + 16 = 25$, so $|v| = 5$.

Lagrangian Connection

The kinetic energy Lagrangian for free motion on a manifold with metric $g$ is:

$L = \frac{1}{2} m\, g(\dot{q}, \dot{q}) = \frac{m}{2}\sum_{ij} g_{ij}(q)\dot{q}^i \dot{q}^j$

This connects geometry to dynamics.

Your Task

Implement metric(G_fn) where G_fn(point) returns the $n \times n$ matrix $[g_{ij}]$. Return a function g such that g(u, v)(*point) computes $g_p(u, v)$.