A properly convex projective manifold is a quotient $\Omega/\Gamma$ where $\Omega\subseteq \mathbb R P^n$ is an open properly convex set and $\Gamma\subseteq PGL(\Omega)$ is discrete and torsion-free.

Definition: Let $M^n$ be a manifold. A projective atlas is a collection $\mathcal A=\{\varphi:U_\alpha\to \mathbb R P^n\}$ such that

$\;\;\;\;(i)$ The sets $U_\alpha\subseteq M$ are open, connected, and cover $M$.

$\;\;\;\;(ii)$ On each component of $U_\alpha \cap U_\beta$ there is a map $g_{\alpha\beta}\in PGL(n+1,\mathbb R)$ with $g_{\alpha\beta}\circ\varphi_\beta=\varphi_\alpha.$

A projective structure is a maximal projective atlas.

Note that a projective structure also defines a smooth structure.

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Local to Global Property

If $g,h\in PGL(n+1,\mathbb R)$ and there exists $U\subseteq \mathbb RP^n$ open such that $g\big\vert_U=h\big\vert_U$ then $g\equiv h$. This rigidity leads us to a notion of developing/analytic continuation as follows.

Let $\varphi_1:U_1\to \mathbb R P^n$ be a chart and fix $p\in U_1.$ Let $\gamma:[a,b]\to M$ such that $\gamma(a)=p$. Cover $\gamma([a,b])$ with charts $U_1,…,U_n.$ We obtain a path $\hat\gamma:[a,b]\to \mathbb R P^n$ given by $$\hat\gamma=(\varphi_1\circ\gamma)(g_{12}\circ\varphi_2\circ\gamma)…(g_{12}\circ g_{23}\circ…\circ g_{(n-1)n}\circ\varphi_n\circ\gamma)$$ where juxtaposition denotes a concatenation of restrictions of paths.

Exercise: The curve $\hat\gamma$ does not depend on the charts (after the first chart $U_1$ is fixed).

Lemma: Let $\alpha,\beta:[a,b]\to M$ with the same endpoints $\alpha(a)=\beta(a),\alpha(b)=\beta(b)$. Let $\varphi:U\to\mathbb RP^n$ be a chart containing $\alpha(a)=\beta(a).$ Suppose that $\alpha$ and $\beta$ are homotopic rel endpoints. Then $\hat \alpha,\hat\beta$ are homotopic rel endpoints.

Proof: If $\alpha([a,b])$ and $\beta([a,b])$ are contained in a simply connected chart then we are done. Let $H:[a,b]^2\to M$ be a homotopy, and pick $% $ such that $H([t_{i-1},t_i]\times[t_{j-1},t_j])$ is contained in a simply connected chart. Using a typical argument, we construct our homotopy from $\hat\alpha$ to $\hat\beta$ piece by piece on all of $[a,b]^2$ using the fact that on each subrectangle, $H([t_{i-1},t_i]\times [t_{j-1},t_j])$ is contained in a simply connected chart. $\blacksquare$

We may now define a “developing map”

$$D:\widetilde M=\{\text{homotopy classes of paths}\}\to \mathbb R P^n$$

by $[\alpha]\mapsto \hat\alpha(b).$ We also obtain a corresponding representation $\rho:\pi_1(M,p)\to PGL(n+1,\mathbb R)$ given by $[\gamma]\mapsto g_{12}g_{23}…g_{(n-1)n}.$ Observe that the developing map is $\rho$-equivariant, i.e. if $\gamma\in \pi_1(M,p)$ and $x\in \widetilde M$ then

$$D(\gamma\cdot x) = \rho(\gamma)D(x)$$

where $\gamma\cdot x$ denotes the action of $\pi_1(M,p)$ on $\widetilde M$ by deck transformations.

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Projective Maps

Now that we have a notion of a projective structure on a manifold, we must determine what constitutes a structure-preserving map between projective manifolds.

Definition: Let $M$ and $N$ be projective manifolds of the same dimension. Let $F:M\to N$ be a continuous map. Then $F$ is a projective map if for all charts $\varphi:U\subset M\to\mathbb RP^n,\psi:V\subset N\to\mathbb RP^n$, given a component of $U\cap F^{-1}(V)$ there exists $g\in PGL(n+1,\mathbb R)$ so that

$$g\circ \varphi = \psi\circ F$$

where both maps above are restricted to the given component of $U\cap F^{-1}(V).$

Exercise: Let $N$ be a projective manifold, let $M$ be a smooth manifold of the same dimension, and let $F:M\to N$ be a local diffeomorphism. Show that there exists a unique projective structure on $M$ for which $F$ is a projective map.

Exercise: Show that $D:\widetilde M \to\mathbb R P^n$ is a projective map.

Exercise: Let $M$ be a simply connected projective manifold. Let $\varphi:U\to \mathbb RP^n$ be a chart. (a) Show that there exists a projective map $F:M\to \mathbb RP^n$ such that $F\big\vert_U=\varphi$. (b) Show that if $F,G:M\to \mathbb RP^n$ are projective maps then there exists a unique $h\in PGL(n+1,\mathbb R)$ such that $F=h\circ G.$

Different choices of charts give different developing maps, but they are related in the following way. Let $D,D’:\widetilde M\to \mathbb RP^n$ be developing maps with holonomies $\rho,\rho’:\pi_1(M)\to PGL(n+1,\mathbb R).$ There exists a unique $g\in PGL(n+1,\mathbb R)$ such that $D’=g\circ D$ and $\rho’=Ad(g)\circ\rho.$ That is, for $\gamma\in \pi_1(M)$

$$\rho’(\gamma)=g\rho(\gamma)g^{-1}.$$

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