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$\DeclareMathOperator\LG{LG}$In the paper The - Conformal geometry of surfaces in the Lagrangian—Grassmannian and second order PDE (published on Proc. London Math. Soc.), I've found an interesting statement:

The Lie quadric $Q^3$, i.e., the space of all points, lines and circles of $\mathbb{R}^2$, is isomorphic to the Lagrangian Grassmannian $\LG(2,4)\subset \operatorname G(2,4)$ of all 2D subspaces of $\mathbb{R}^4$ which are isotropic with respect to the canonical symplectic form $\eta=e^1\wedge e^3+e^2\wedge e^4$, where $\{e^i\}$ is a dual basis of $\mathbb{R}^4$.

But this fact is not proved clearly, and I'm beginning to wonder whether it is actually true.

Just looking at the dimensions, everything looks all-right: $\LG(2,4)$ contains an open subset of symmetric $2\times 2$ matrices, so it must be 3D; to parameterise all circles in $\mathbb{R}^2$ (comprising zero and infinite radii) I need 3 numbers (2 for the position of the center and 1 for the radius).

Then, the actual proof should go as follows: Plücker-embed $\LG(2,4)$ into $\mathbb{P}(\ker \eta)\cong\mathbb{RP}^4$, where $\ker \eta\subset{\bigwedge}^2\mathbb{R}^4$, and observe that its image is the 3D quadric $Q\subset \mathbb{RP}^4$ of isotropic elements with respect to the obviously defined (conformal) symmetric form $$ {\bigwedge}^2\mathbb{R}^4\otimes{\bigwedge}^2\mathbb{R}^4\longrightarrow {\bigwedge}^4\mathbb{R}^4\cong\mathbb{R}. $$ On the other hand, $Q^3$ is the 3D quadric of isotropic lines in $\mathbb{R}^{2,3}$, but I cannot manage to make the last step and answer myself the following:

QUESTION: Is it true that $Q=Q^3$?

In the affirmative case, it should be such a basic fact that even several alternative proofs should have be given in literature—but I wasn't able to find any. For example, I'd like to see a very elementary one, i.e., not involving multilinear algebra, but just incidence/tangency of lines, planes and circles. Any reference will be appreciated!

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  • $\begingroup$ Thanks for linking to the freely available copy (as well as pointing out the 'official' version). $\endgroup$
    – David Roberts
    May 12, 2014 at 10:33

2 Answers 2

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This, of course, is "classical". Here is a simple way to see the identification between the space of all circles (including point circles) on the two-sphere and the Lagrangian Grassmannian in $\mathbb{R}^4$ taken from Section 5 of the paper Finsler surfaces with prescribed geodesics by Gautier Berck and myself.

Consider $S^3$ as the set unit quaternions and let $\pi : S^3 \rightarrow S^2$ be the Hopf fibration $\pi(q) := qi\bar{q}$. Every Lagrangian subspace in $\mathbb{R}^4$ intersects the sphere in a (Legendrian) great circle and the image of this great circle under the Hopf fibration is a circle on the two-sphere. Conversely, every circle on the two-sphere can be "lifted" to a Legendrian great circle on the three-sphere (the lift is actually the Frenet frame of the circle in the group of unit quaternions or, equivalently, in $SU(2)$).

The details are in the paper.

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    $\begingroup$ I define something "classical" as something that when you see stated, you say to yourself "I should have known that", then you ask around and the people you ask tell you that "it must be well known" but cannot point you to a reference. You then ask some very educated mathematicians and they know it, but they give you really complicated explanations. Finally you think about it, do it yourself, and when you explain it to others they all tell you they knew that. $\endgroup$ May 10, 2014 at 8:28
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    $\begingroup$ I don't know the exact reference, but this fact is in Lie's work. He proved that the group of contact transformations of the plane that carry circles to circles (points are 'ideal circles' in the classical language) is a $10$-dimensional group isomorphic to what we now call $\mathrm{SO}(2,3)$, which is, in turn, 'isomorphic' to what we now call $\mathrm{Sp}(2,\mathbb{R})$. Of course, he didn't make a careful distinction between real and complex, and he didn't worry too much about 'points at infinity' (i.e., global questions). I'm sure I could find the exact reference, if you want to know it. $\endgroup$ May 10, 2014 at 9:53
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    $\begingroup$ Lie's thesis (as translated in Smith's "A Source Book in Mathematics") already has this, if my memory serves me right. $\endgroup$ May 10, 2014 at 10:24
  • $\begingroup$ @alvarezpaiva your explanation is "almost" what I needed. I can see that intersecting $S^3$ with a Lagrangian plane yields a circle on $S^2$ via Hopf fibration, but then - I guess - it still needs to be stereographically projected to $\mathbb{R}^2$ to become an actual circle on a plane. Still, I don't see where 'ideal circles' come from (I think your construction cannot deal with them: what is the Frenet frame of a point?) But I have to read carefully your paper, and Lie's work, kindly suggested by R. Bryant, and will take a few days... meanwhile, I'll be waiting for more hints from MO users! $\endgroup$ May 12, 2014 at 7:35
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    $\begingroup$ @G_infinity: Since the details are in the paper I mentioned, I just gave a (complete) sketch of the construction. You can of course switch to the plane if you want to by stereographic projection. Point circles are dealt with automatically because they correspond to the fibers of the Hopf fibration, which are intersections of the three-sphere with Lagrangian subspaces (provided the symplectic structure or the Hopf fibration are chosen "correctly"). $\endgroup$ May 12, 2014 at 10:27
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Replying to this somewhat late but I think there is important context to this. The Lagrangian Grassmannian and the quadric are both examples of R-spaces (symmetric R-spaces in particular) as such they are given by $ G/P $ for $G$ a semisimple Lie Group and $P$ some parabolic subgroup. So this question boils down to $ \mathfrak{sp}(4,\mathbb{R}) \cong \mathfrak{so}(2,3) $ which is one of the exceptional Lie isomorphisms. (Note that the stabiliser of an isotropic plane in $\mathbb{R}^4$ is sent to that of a null line in $\mathbb{R}^{2,3}$.) All the projective quadrics up to dimension 4 have such an identification to some Grassmannian (isotropic, isotropic for a hermitian form, Lagrangian or normal) over the reals, complex numbers or quaternions.

From a more practical perspective take the Klein correspondence: $G_2(\mathbb{R}^4)$ embeds into $\mathbb{P}(\bigwedge^2 \mathbb{R}^4)$ as the light cone. Then a symplectic form $\omega$ on $\mathbb{R}^4$ is an element of $\bigwedge^2 (\mathbb{R}^4)^* \cong \left(\bigwedge^2 \mathbb{R}^4\right)^*$. A plane $V$ is isotropic iff $\bigwedge^2 V$ is in the annihilator of $\omega$. Then if $\omega$ is nondegenerate the annihilator must have signature $(2,3)$ or $(3,2)$ so its intersection with the Klein quadric is the Lie quadric you require.

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