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 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

Not to be confused with the plane in hyperbolic geometry.

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.

See also

• Isotropic line
• Polar space
• Witt group
• Witt ring (forms)
• Universal quadratic form

References

1. 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

• 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.