hyperbolic plane

The hyperbolic plane is the simplest nontrivial ‘mixed-sign’ geometry: a two-dimensional space carrying a quadratic form with a built-in pair of light-like directions. It is the universal building block of indefinite forms — over any field, every nondegenerate isotropic form contains one, and stripping them away leaves an anisotropic core. (This is a form-theoretic object; it is unrelated to the hyperbolic plane of non-Euclidean geometry that shares the name.)

Concretely, the hyperbolic plane H over a field K is the two-dimensional quadratic space with a basis e, f satisfying Q(e) = Q(f) = 0 and B(e, f) = 1; in the basis (e, f) the form is Q(x e + y f) = 2xy, with Gram matrix [0, 1; 1, 0]. Equivalently, after the change u = x + y, v = x − y it is x'^2 − y'^2 (up to scaling), the standard split form of signature (1, 1) over R. It is nondegenerate, isotropic, and has Witt index 1.

Hyperbolic planes are the ‘zero’ of the Witt ring: H represents the trivial class, and a form is hyperbolic (an orthogonal sum of copies of H) exactly when it dies in W(K). By Witt decomposition any nondegenerate form is uniquely an orthogonal sum of some number r of hyperbolic planes — r being the Witt index — and an anisotropic part. Thus the hyperbolic planes carry all and only the isotropic content of a form.