Isotropic quadratic form

Last updated December 02, 2022

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 of V. 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.

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. An important example of an isotropic form over the reals occurs in pseudo-Euclidean space.

Hyperbolic plane

Let F be a field of characteristic not 2 and V = F². If we consider the general element (x, y) of V, then the quadratic forms q = xy and r = x² − y² 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 Husemoller^[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 M² = N² = 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.

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References

1 2 3 4 5 Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.
↑ Emil Artin (1957) Geometric Algebra, page 119 via Internet Archive

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. Vol. 7 (reprint of 3rd ed.). Springer-Verlag. ISBN 0-387-90040-3. Zbl 1034.11003.

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[MH-1] 1 2 3 4 5 Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.

[2] Emil Artin (1957) Geometric Algebra, page 119 via Internet Archive

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