A properly convex projective manifold is a quotient where is an open properly convex set and is discrete and torsion-free.

Definition: Let be a manifold. A projective atlas is a collection such that

The sets are open, connected, and cover .

On each component of there is a map with

A projective structure is a maximal projective atlas.

Note that a projective structure also defines a smooth structure.


Local to Global Property

If and there exists open such that then . This rigidity leads us to a notion of developing/analytic continuation as follows.

Let be a chart and fix Let such that . Cover with charts We obtain a path given by \begin{equation} \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) \end{equation} where juxtaposition denotes a concatenation of restrictions of paths.

Exercise: The curve does not depend on the charts (after the first chart is fixed).

Lemma: Let with the same endpoints . Let be a chart containing Suppose that and are homotopic rel endpoints. Then are homotopic rel endpoints.

Proof: If and are contained in a simply connected chart then we are done. Let be a homotopy, and pick such that is contained in a simply connected chart. Using a typical argument, we construct our homotopy from to piece by piece on all of using the fact that on each subrectangle, is contained in a simply connected chart.

We may now define a “developing map”

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

by We also obtain a corresponding representation given by Observe that the developing map is -equivariant, i.e. if and then

\begin{equation} D(\gamma\cdot x) = \rho(\gamma)D(x) \end{equation}

where denotes the action of on by deck transformations.


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 and be projective manifolds of the same dimension. Let be a continuous map. Then is a projective map if for all charts , given a component of there exists so that

\begin{equation} g\circ \varphi = \psi\circ F \end{equation}

where both maps above are restricted to the given component of

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

Exercise: Show that is a projective map.

Exercise: Let be a simply connected projective manifold. Let be a chart. (a) Show that there exists a projective map such that . (b) Show that if are projective maps then there exists a unique such that

Different choices of charts give different developing maps, but they are related in the following way. Let be developing maps with holonomies There exists a unique such that and That is, for

\begin{equation} \rho’(\gamma)=g\rho(\gamma)g^{-1}. \end{equation}

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