Lowering an index with the metric

The metric is the bridge between vectors and covectors. Without it, V_p and V_p* are separate vector spaces with the same dimension but no canonical identification. A metric g picks out one specific isomorphism: v ↔ ω where ω(w) := g(w, v) for all w. The covector ω appears here as a stack of integer level lines (drawn from covector.mjs); raising the index again with g⁻¹ recovers v.

Read the picture: with the Euclidean metric, the level lines are perpendicular to v — the usual “projection onto a unit vector” intuition. Stretch λ₁: the metric is no longer isotropic, the level lines tilt and crowd, and the orange g(v,v)=1 curve becomes an ellipse. Push λ₂ below zero (or pick the Lorentzian preset): the curve splits into a hyperbola and lightlike v (along the asymptotes) gets sent to the zero covector. This is exactly the picture in §6.7 of the notes — raising and lowering indices is geometry, not just algebra.