Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I am always fascinated when a quadratic form (or a quadric) arises naturally. I have some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too vague for MO. Most forms I list are really elementary, and all are finite dimensional.

I got most of the following examples from M.Berger, Geometry I & II, and from the truly beautiful book "Eléments de géométrie : actions de groupes" by french author Rached Meinmné.

$(0)$ the discriminant on the affine space of unitary degre 2 polynomials

$(i)$ the determinant on endomorphisms of a 2 dimensional vector space, and $\mathrm{Tr}^2-4\mathrm{det}$

$(ii)$ the radical on the space of quadratic forms on a 2 dimensional vector space, and the isotrope cone (not sure about the name, degenerate cone?).

$(iii)$ the family of hermitian forms (built from the Wronskian) on the solution space of the discrete Schroedinger equation that allow one to show the existence of right and left side $L^2$ solutions, and the Weyl m function.

$(iv)$ If $\Delta$ is any $2$ dimensional complex vector space, then $\mathrm{Herm}(\Delta)$, the real vector space of hermitian forms on $\Delta$, carries a natural quadratic form obtained by constructing an essentially unique morphism $\rho$ from $\mathrm{Herm}(\Delta)$ to $\mathrm{Hom}(\Delta\oplus\overline{\Delta})$ such that for all $h\in\mathrm{Herm}(\Delta),~\rho(h)^2$ is proportional to $\mathrm{Id}$, the proportionality defining the quadratic form. Here, $\rho$ only depends on a choice of a nonzero element $\omega\in\Lambda^2\Delta^*$.

$(v)$ If $V$ is a 4 dimensional vector space, then $\Lambda^2 V$ carries the natural quadric $Q(v)=v\wedge v$ where $\Lambda^4 V$ is identified with the underlying field, which vanishes exactly when $v$ comes from the canonical map $\mathrm{Gr}(2,V)\rightarrow P\Lambda^2V$.

I remember reading about one on the space of circles, but I forgot the details. What other examples of natural quadratic forms are there?

share|improve this question
Second-order approximation to a potential function at a critical point? –  Qiaochu Yuan Mar 18 '11 at 20:27
I think the question is a little broad. Can you be more specific about what you are looking to gain by having such a list? –  Qiaochu Yuan Mar 18 '11 at 20:34
People are quite fond of : $$ $$ Symmetric bilinear forms [by] J. Milnor [and] D. Husemoller by John Willard Milnor, 1973,Springer-Verlag edition, in English. –  Will Jagy Mar 18 '11 at 21:10
What do you mean by "arise naturally"? In your examples you are often introducing constraints on a dimension to force a quadratic form in what otherwise would really be better considered as a homogeneous polynomial of some degree. For instance, the determinant on endomorphisms of an n-dim. space is a homogeneous polynomial of degree n in n variables. To say the determinant becomes a quadratic form when you set n = 2 seems, to me, to be missing the big picture of what happens in general, since the determinant is usually not a quadratic form. (Continued...) –  KConrad Mar 18 '11 at 22:11
I would consider as examples that give incentive to study general quadratic forms those constructions which, in general, produce quadratic forms in potentially any number of variables, not just a small number because you fixed some parameter to force a quadratic form. For instance, the trace form on an associative algebra or the Killing form on a Lie algebra. These are symmetric bilinear forms in possibly many variables (depends on the dimension of the algebra) and correspond to some quadratic forms in many variables. –  KConrad Mar 18 '11 at 22:14

2 Answers 2

Dear Olivier, in line with the more advanced nature of this site, let me give an example of a less elementary nature.

Consider a compact Riemann surface $X$ of genus 2 and on it stable vector bundles $E$ of rank 2 whose determinant bundle $\Lambda ^2E$ is isomorphic to some fixed line bundle $L$ of degree $-1$. Newstead has proved that the moduli space of those vector bundles is the intersection of two quadrics in five-dimesional projective space $\mathbb P^5(\mathbb C)$. And one of those quadrics is the Klein quadric in $\mathbb P^5(\mathbb C)$ parametrizing the lines in some three-dimensional projective space canonically associated to $X$ and $L$. (A Klein quadric is the quadric you mention in number (v) of your list.)

P E. Newstead Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology 7 (1968), 205-215.
For a geometric description including the role of the Klein quadric, see:
M. S. Narasimhan and S. Ramanan Moduli of Vector Bundles on a Compact Riemann Surface, Annals of Mathematics, Vol. 89, No. 1, 1969 , pp. 14-51.

share|improve this answer
@Olivier: when you ask such a broad, open-ended question, it seems to be good form to wait more than a couple of hours to accept an answer. –  Pete L. Clark Mar 19 '11 at 5:55
you're right, what I meant to do is to give his answer a thumbs up, not close the question. –  Olivier Bégassat Mar 20 '11 at 0:03

Perhaps too trivial an example; the fixed points of a (non-intentity) Mobius transform.

share|improve this answer
can you explain please? I looked up "fixed points of Moebius transforms" and only found that (the non identity ones) they either fix two points on the boundary $\partial\mathbb{H}$, one point on the boundary and no other, or two points in $\mathbb{H}$ –  Olivier Bégassat Mar 20 '11 at 0:28

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.