Lecture 3: Properly Convex Domains

Last time: $\theta\colon PSL(2,R)\hookrightarrow SL(3,R)$. If $V$ is a real vector space, let $S^2(V)$ be the collection of quadratic forms $\beta\colon V\to R$. These are functions of the form $\ell x^2+2mxy+ny^2$ for real numbers $\ell,m,n$. This can be expressed as a symmetric matrix $\beta=\begin{pmatrix} \ell & m \\ m & n\end{pmatrix}$. There is an action of $SL(2,R)$ on $S^2(R^2)$, given by, for $A\in SL(2,R)$, setting $\theta(A)\colon S^2(R^2)\to S^2(R^2)$ to be $(A^{-1})^t\begin{pmatrix} \ell & m \\ m & n\end{pmatrix}A^{-1}$. Concretely, if $A=\begin{pmatrix}a & b \\ c & d\end{pmatrix}$, we have $\theta(A)\beta=\begin{pmatrix}d^2\ell -2dcm +c^2n & -db\ell + (ad+bc)m-acn \\ -db\ell +(ad+bc)m-acn & b^2\ell-2abm+a^2n\end{pmatrix}$.

By identifying $S^2(\mathbb{R}^2)$ with $\mathbb{R}^3$ via $\beta\mapsto (\ell,m,n)$, this gives $\theta(A)=\begin{pmatrix}d^2 & -2dc & c^2 \\ -2b\ell & ad+bc & -2ac \\ b^2 & -2ab & a^2\end{pmatrix}$.

Claim: $\text{Im}(\theta)=SO(1,2)=\text{Isom}_+(J)$, where $J=\begin{pmatrix} 0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0\end{pmatrix}$.

This is because the determinant map which sends $\beta$ to $\ell n-m^2$ is invariant under the action of $SL(2,R)$, namely since $\theta(A)(\beta)=(A^{-1})^tP(A^{-1})$, and $\det(A)=1$, we have $\det(\theta(A)(\beta))=\det(\beta)$. Therefore, the image of $\theta$ is a subspace of $\text{Isom}_+(J)$, and by comparing dimensions and using the fact that $\text{Isom}_+(J)$ is connected, we get equality.

Fact: $SL(2,R)/SO(2)\cong H^2$. Therefore $\text{Im}(\theta)/\theta(SO(2))=SO(1,2)/\theta(SO(2))\cong H^2$.

$SO(1,2)$ acting on $R^3$. By mapping $(\ell,m,n)\mapsto(x_3-x_1,x_2,x_1+x_3)$, we change $\ell n-m^2$ to $x_3^2-x_1^2-x_2^2$. Given an $M\in SO(2,1)$, this acts on the new coordinates by conjugation with the coordinate map, and preserves the quadratic form. In particular, the 1-locus is a hyperboloid of two sheets which is preserved under the action of $SO(2,1)$, and we can metrize the positive (meaning $x_3>0$) sheet of this hyperboloid $H$ with a hyperbolic metric.

So the action of $SO(1,2)$ on $H$ is conjugate by a diffeomorphism to action of $PSL(2,R)$ on $H^2=\{x+iy\mid y>0\}$.

We may also projectivize $R^3$, in which case both sheets of the hyperboloid are identified with the unit disc $\mathbb{D}$ in the affine patch $[x_1:x_2:1]$, with $x_1^2+x_2^1<1$. Then the action of $PSO(1,2)$ on $\mathbb{D}$ is conjugate by a diffeomorphism to an action of $PSL(2,R)$ on the upper half plane $H^2$. Then $(\mathbb{D},PO(1,2))$ is called the Klein model of $H^2$.

Definition: A subset $\Omega\subseteq R P^n$ is convex if for every line $\ell$ in $R P^n$, $\Omega \cap \ell$ is connected (possibly empty). If also the closure $\bar{\Omega}$ is disjoint from some $R P^{n-1}$, then $\Omega$ is properly convex. For example, consider $\mathbb{D}\subseteq R P^2$.

Definition: A properly convex projective orbifold is $M^n=\Omega/\Gamma$, where $\Omega$ is a properly convex open set in $R P^n$ and $\Gamma$ is a discrete subgroup of $PGL(\Omega)=\{\alpha\in PGL(n+1,R)\mid \alpha(\Omega)=\Omega\}$. If $\Gamma$ has no elements of finite order, or if $\Gamma$ acts freely on $\Omega$, this is a manifold.

Example: A hyperbolic manifold is a properly convex projective orbifold.

By taking $\Omega$ to be the interior of a triangle, and $\Gamma =\langle \begin{pmatrix} 2 & & \\ & 1 & \\ & & 1\end{pmatrix},\begin{pmatrix} 1 & & \\ &2 & \\ & & 1\end{pmatrix}\rangle$, we obtain a torus.

Duality: Let $V^*=\text{Hom}(V,R)$, and consider $P(V^*)$. A $T\in GL(V)$ acts on $V^*$ by $T_*(\phi)=\phi\circ T^{-1}$.

If $\beta$ is a basis of $V$, and $\beta^*$ is the dual basis of $V^*$, then the matrices are related by $[T_*]_{\beta^*}=([T]_\beta^{-1})^t$. The automorphism of $GL(n,\mathbb{R})$ given by $A\mapsto (A^{-1})^t$ is called a global Cartan involution.

Let $\Omega$ be a subset of an affine patch $A^n$ in $R P^n$. Define $\mathscr{C}(\Omega)=\{tv\mid v\in \Omega, t>0\}$. For example, if $\Omega=\mathbb{D}$, then $\mathscr{C}\mathbb{D}$ is the interior of the light cone, the set $\{(x_1,x_2,x_3)\mid x_3^2>x_1^2+x_2^2,x_3>0\}$.

Let $\mathscr{C}^*\Omega=\{\phi\in V^*\mid \phi(x)>0\forall x\in\overline{\mathscr{C}\Omega}\}$. This is an open convex cone, because if $\phi_1,\phi_2\in\mathscr{C}^*$ and $\lambda\in [0,1]$, then $(\lambda\phi_1+(1-\lambda)\phi_2)(x)>0$ for all $x\in \overline{\mathscr{C}\Omega}$.

Definition: The dual domain of $\Omega$ is $\Omega^*=P(\mathscr{C}^*\Omega)$, which is a convex set in $P(V^*)$. It is open and properly convex.

Example: Take $\Omega$ to be the image of a spherical triangle in $S^2$ in $R P^2$, with angles $\alpha,\beta,\gamma$ and opposing side lengths $a,b,c$. The standard inner product of $V=R^3$ gives a canonical isomorphism from $V\to V^*$ by mapping $v$ to $\langle v,-\rangle$ (linear functionals are representable). This identifies $P(V)$ with $P(V^*)$, and in this case, $\Omega^*$ is a triangle with side lengths $\alpha,\beta,\gamma$ and angles $a,b,c$.

$\Omega$ is strictly convex if it is properly convex and its boundary contains no line segments. For example, a triangle is properly but not strictly convex.

Previous Post: Lecture 2: Introduction to Projective Geometry Part II

Next Post: Lecture 4: Projective Structures and Maps