hyperbolic plane in quadratic spaces
A non-singular (http://planetmath.org/NonDegenerateQuadraticForm) isotropic quadratic space of dimension
2 (over a field) is called a hyperbolic plane. In other
words, is a 2-dimensional vector space
over a field
equipped with a quadratic form
such that there exists a non-zero
vector with .
Examples. Fix the ground field to be , and be the two-dimensional vector space over with the standard basis and .
-
1.
Let . Then for all . is a hyperbolic plane. When is written in matrix form, we have
-
2.
Let . Then for all . is a hyperbolic plane. As above, 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 hyperbola in a two-dimensional Euclidean plane
.
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 and , or in matrix form:
In fact, we have the following
Proposition. Any two hyperbolic planes over a field of
characteristic not 2 are isometric quadratic spaces.
Proof.
From the first example above, we see that the quadratic space with the quadratic form is a hyperbolic plane. Conversely, if we can show that any hyperbolic plane is isometric the example (with the ground field switched from to ), we are done.
Pick a non-zero vector and suppose it is
isotropic: . Pick another vector so
forms a basis for . Let be
the symmetric bilinear form associated with . If ,
then for any with , , contradicting the fact that
is non-singular. So . By dividing
by , we may assume that .
Suppose . Then the matrix associated with the quadratic form corresponding to the basis is
If then we are done, since is equivalent to via the isometry
given by
If , then the trick is to replace with an isotropic vector so that the bottom right cell is also 0. Let . It’s easy to verify that . As a result, the isometry required has the matrix form
∎
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 or . Its notation, corresponding to the second of the forms, is , or simply .
A hyperbolic space is a finite dimensional orthogonal direct sum of hyperbolic planes. It is always even dimensional and has the notation or simply , where is the dimensional of the hyperbolic space.
Remarks.
-
•
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 curvature
(Euclidean
signature
) that is commonly used in differential geometry
, 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 |