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.

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