hyperbolic plane in quadratic spaces

A non-singularPlanetmathPlanetmath (http://planetmath.org/NonDegenerateQuadraticForm) isotropic quadratic space of dimensionPlanetmathPlanetmath 2 (over a field) is called a hyperbolic plane. In other words, is a 2-dimensional vector spaceMathworldPlanetmath over a field equipped with a quadratic formMathworldPlanetmath Q such that there exists a non-zero vector v with Q(v)=0.

Examples. Fix the ground field to be , and 2 be the two-dimensional vector space over with the standard basis (0,1) and (1,0).

  1. 1.

    Let Q1(x,y)=xy. Then Q1(a,0)=Q1(0,b)=0 for all a,b. (2,Q1) is a hyperbolic plane. When Q1 is written in matrix form, we have


  2. 2.

    Let Q2(r,s)=r2-s2. Then Q2(a,a)=0 for all a. (2,Q2) is a hyperbolic plane. As above, Q2 can be written in matrix form:


From the above examples, we see that the name “hyperbolic plane” comes from the fact that the associated quadratic form resembles the equation of a hyperbolaMathworldPlanetmathPlanetmath in a two-dimensional Euclidean planeMathworldPlanetmath.

It’s not hard to see that the two examples above are equivalent quadratic forms. To transform from the first form to the second, for instance, follow the linear substitutions x=r-s and y=r+s, or in matrix form:


In fact, we have the following

PropositionPlanetmathPlanetmath. Any two hyperbolic planes over a field k of characteristic not 2 are isometric quadratic spaces.


From the first example above, we see that the quadratic space with the quadratic form xy is a hyperbolic plane. Conversely, if we can show that any hyperbolic plane is isometric the example (with the ground field switched from to k), we are done.

Pick a non-zero vector u and suppose it is isotropic: Q(u)=0. Pick another vector v so {u,v} forms a basis for . Let B be the symmetric bilinear formMathworldPlanetmath associated with Q. If B(u,v)=0, then for any w with w=αu+βv, B(u,w)=αB(u,u)+βB(u,v)=0, contradicting the fact that is non-singular. So B(u,v)0. By dividing v by B(u,v), we may assume that B(u,v)=1.

Suppose α=B(v,v). Then the matrix associated with the quadratic form Q corresponding to the basis 𝔟={u,v} is


If α=0 then we are done, since M𝔟(Q) is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to M𝔟(Q1) via the isometryMathworldPlanetmath T: given by

T=(12001), so that Tt(0110)T=(012120).

If α0, then the trick is to replace v with an isotropic vector w so that the bottom right cell is also 0. Let w=-α2u+v. It’s easy to verify that Q(w)=0. As a result, the isometry S required has the matrix form

S=(1-α201), so that St(011α)S=(0110).

Thus we may speak of the hyperbolic plane over a field without any ambiguity, and we may identify the hyperbolic plane with either of the two quadratic forms xy or x2-y2. Its notation, corresponding to the second of the forms, is 1-1, or simply 1,-1.

A hyperbolic space is a finite dimensional orthogonal direct sum of hyperbolic planes. It is always even dimensional and has the notation 1,-1,1,-1,,1,-1 or simply n1n-1, where 2n is the dimensional of the hyperbolic space.


  • The notion of the hyperbolic plane encountered in the theory of quadratic forms is different from the “hyperbolic plane”, a 2-dimensional space of constant negative curvaturePlanetmathPlanetmath (EuclideanPlanetmathPlanetmath signaturePlanetmathPlanetmathPlanetmathPlanetmath) that is commonly used in differential geometryMathworldPlanetmath, and in non-Euclidean geometry.

  • Instead of being associated with a quadratic form, a hyperbolic plane is sometimes defined in terms of an alternating form. In any case, the two definitions of a hyperbolic plane coincide if the ground field has characteristic 2.

Title hyperbolic plane in quadratic spaces
Canonical name HyperbolicPlaneInQuadraticSpaces
Date of creation 2013-03-22 15:41:47
Last modified on 2013-03-22 15:41:47
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 11E88
Classification msc 15A63
Defines hyperbolic plane
Defines hyperbolic space