# Isotropic quadratic form

In mathematics, a quadratic form over a field *F* is said to be **isotropic** if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is **anisotropic**. More precisely, if *q* is a quadratic form on a vector space *V* over *F*, then a non-zero vector *v* in *V* is said to be **isotropic** if *q*(*v*) = 0. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form.

Suppose that (*V*, *q*) is quadratic space and *W* is a subspace. Then *W* is called an **isotropic subspace** of *V* if *some* vector in it is isotropic, a **totally isotropic subspace** if *all* vectors in it are isotropic, and an **anisotropic subspace** if it does not contain *any* (non-zero) isotropic vectors. The **isotropy index** of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces.[1]

A quadratic form *q* on a finite-dimensional real vector space *V* is anisotropic if and only if *q* is a definite form:

- either
*q*is*positive definite*, i.e.*q*(*v*) > 0 for all non-zero*v*in*V*; - or
*q*is*negative definite*, i.e.*q*(*v*) < 0 for all non-zero*v*in*V*.

- either

More generally, if the quadratic form is non-degenerate and has the signature (*a*, *b*), then its isotropy index is the minimum of *a* and *b*.

## Hyperbolic plane

Let *F* be a field of characteristic not 2 and *V* = *F*^{2}. If we consider the general element (*x*, *y*) of *V*, then the quadratic forms *q* = *xy* and *r* = *x*^{2} − *y*^{2} are equivalent since there is a linear transformation on *V* that makes *q* look like *r*, and vice versa. Evidently, (*V*, *q*) and (*V*, *r*) are isotropic. This example is called the **hyperbolic plane** in the theory of quadratic forms. A common instance has *F* = real numbers in which case {*x* ∈ *V* : *q*(*x*) = nonzero constant} and {*x* ∈ *V* : *r*(*x*) = nonzero constant} are hyperbolas. In particular, {*x* ∈ *V* : *r*(*x*) = 1} is the unit hyperbola. The notation ⟨1⟩ ⊕ ⟨−1⟩ has been used by Milnor and Huseman[1]^{:9} for the hyperbolic plane as the signs of the terms of the bivariate polynomial *r* are exhibited.

The affine hyperbolic plane was described by Emil Artin as a quadratic space with basis {*M*, *N*} satisfying *M*^{2} = *N*^{2} = 0, *NM* = 1, where the products represent the quadratic form.[2]

Through the polarization identity the quadratic form is related to a symmetric bilinear form *B*(*u*, *v*) = 1/4(*q*(*u* + *v*) − *q*(*u* − *v*)).

Two vectors *u* and *v* are orthogonal when *B*(*u*, *v*) = 0. In the case of the hyperbolic plane, such *u* and *v* are hyperbolic-orthogonal.

## Split quadratic space

A space with quadratic form is **split** (or **metabolic**) if there is a subspace which is equal to its own orthogonal complement; equivalently, the index of isotropy is equal to half the dimension.[1]^{:57} The hyperbolic plane is an example, and over a field of characteristic not equal to 2, every split space is a direct sum of hyperbolic planes.[1]^{:12,3}

## Relation with classification of quadratic forms

From the point of view of classification of quadratic forms, anisotropic spaces are the basic building blocks for quadratic spaces of arbitrary dimensions. For a general field *F*, classification of anisotropic quadratic forms is a nontrivial problem. By contrast, the isotropic forms are usually much easier to handle. By Witt's decomposition theorem, every inner product space over a field is an orthogonal direct sum of a split space and an anisotropic space.[1]^{:56}

## Field theory

- If
*F*is an algebraically closed field, for example, the field of complex numbers, and (*V*,*q*) is a quadratic space of dimension at least two, then it is isotropic. - If
*F*is a finite field and (*V*,*q*) is a quadratic space of dimension at least three, then it is isotropic (this is a consequence of the Chevalley-Warning theorem). - If
*F*is the field*Q*_{p}of*p*-adic numbers and (*V*,*q*) is a quadratic space of dimension at least five, then it is isotropic.

## References

- Milnor, J.; Husemoller, D. (1973).
*Symmetric Bilinear Forms*. Ergebnisse der Mathematik und ihrer Grenzgebiete.**73**. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016. - Emil Artin (1957) Geometric Algebra, page 119

- Pete L. Clark, Quadratic forms chapter I: Witts theory from University of Miami in Coral Gables, Florida.
- Tsit Yuen Lam (1973)
*Algebraic Theory of Quadratic Forms*, §1.3 Hyperbolic plane and hyperbolic spaces, W. A. Benjamin. - Tsit Yuen Lam (2005)
*Introduction to Quadratic Forms over Fields*, American Mathematical Society ISBN 0-8218-1095-2 . - O'Meara, O.T (1963).
*Introduction to Quadratic Forms*. Springer-Verlag. p. 94 §42D Isotropy. ISBN 3-540-66564-1. - Serre, Jean-Pierre (2000) [1973].
*A Course in Arithmetic*. Graduate Texts in Mathematics: Classics in mathematics.**7**(reprint of 3rd ed.). Springer-Verlag. ISBN 0-387-90040-3. Zbl 1034.11003.