Signature shape-shifter

Sylvester's law of inertia in one picture. The geometry of g is captured by its signature — the number of positive versus negative eigenvalues. In 2D there are essentially three cases: (+,+) Riemannian (the unit ball is an ellipse, distances are positive, angles make sense in the usual way); (+,−) Lorentzian (the “unit” locus is a hyperbola with two branches, lightlike vectors lie on the asymptotes & have g(v,v) = 0); (−,−) negative definite (no real v satisfies g(v,v) = 1, hence empty locus).

Drag λ₂ from +1 down through 0 and into negative territory and watch the curve morph: ellipse stretches, becomes a pair of parallel lines at λ₂ = 0, then opens up into a hyperbola. The degenerate “parallel lines” case shows what goes wrong when nondegeneracy fails — this is why the metric is required to be nondegenerate in the definition.