# Lecture 14: Deforming Strictly Convex Projective Manifolds

Let \(M\) be a closed, connected, smooth \(n\)-manifold.

Our goal is to describe all strictly convex projective structures on \(M\). To this end, we seek a strictly convex domain \(\Omega \subseteq \mathbb R \mathrm P^n\) such that \(M\) is homeomorphic to the orbit space \(\Omega / \Gamma\) for some subgroup \(\Gamma\) of \(\mathrm{PGL}(\Omega)\). Recall that \(\Omega\) is *properly convex* if its closure lies in an affine patch, and that \(\Omega\) is *strictly convex* if it is properly convex and if \(\partial \overline \Omega\) contains no line segment of positive length.

A marking \(h : M \to \Omega / \Gamma\) gives a *holonomy representation*

**Proposition.** *\(\Gamma\) determines \(\Omega\); hence, \(\rho\) determines \(\Omega\).*

*Proof.* This follows from the fact that \(\partial \overline \Omega\) is the limit set of \(\Gamma\). \(\square\)

Thus describing strictly convex projective structures on \(M\) amounts to describing holonomy representations \(\pi_1 M \to \Gamma \leq \mathrm{PGL}(n + 1)\).

Let \(SC(M)\) denote the set of all holonomy representations \(\pi_1 M \to \mathrm{PGL}(n + 1)\) of strictly convex projective structures on \(M\). We have

\[SC(M) \leq \mathrm{Hom}(\pi_1 M, \mathrm{PGL}(n + 1)) \cong \mathrm{Hom}(\pi_1 M, \mathrm{SL}_\pm (n + 1)).\]**Theorem** (Benoist 2000)**.** *\(SC(M)\) is both open and closed in \(\mathrm{Hom}(\pi_1 M, \mathrm{PGL}(n + 1))\).*

We will prove the “open” part of this theorem using a general result of Thurston and Ehresmann.

*\((G, X)\)-MANIFOLDS, THE HOLONOMY THEOREM*

Let \(X\) be a manifold, and let \(G\) be a Lie group that acts continuously (and hence analytically) and transitively on \(X\) by diffeomorphisms.

Recall that a manifold \(M\) is a *\((G, X)\)-manifold* if there exists an atlas \(\{\varphi_i : U_i \subseteq M \to \varphi_i(U_i) \subseteq X\}\) of \(M\) such that the transition map

is the restriction of an element of \(G\) whenever \(U_i \cap U_j\) is nonempty, and such that the following *cocycle condition* is satisfied:

Alternatively, \(M\) is a \((G, X)\)-manifold if and only if it is homeomorphic to a quotient \(X / \Gamma\) for some discrete torsion-free subgroup of \(G\). Then the action of \(\Gamma\) on \(X\) is properly discontinuous and free, and the diagram

\[\begin{matrix} \pi_1 M & \smash{\stackrel{\mathrm{hol}}{\longrightarrow}} & \Gamma \\ \curvearrowright & & \curvearrowright \\ \widetilde M & \smash{\stackrel{\mathrm{dev}}{\longrightarrow}} & X \\ \end{matrix}\]commutes. Here \(\pi_1 M\) acts on the universal cover \(\widetilde M\) of \(M\) by deck transformations, \(\mathrm{dev}\) is the developing map, and \(\mathrm{hol}\) is the holonomy of \(M\).

**Theorem** (Thurston–Ehresmann)**.** *Let \(M\) be a closed, connected, smooth \((G, X)\)-manifold. Then the set of all holonomies \(\pi_1 M \to G\) of \((G, X)\)-structures on \(M\) is open in \(\mathrm{Hom}(\pi_1 M, G)\).*

*Sketch of proof* (Thurston 1978)*.* Let \(\pi : \widetilde M \to M\) be the universal cover, and let \(D \subseteq \widetilde M\) be a fundamental domain. There exist finitely many charts fo \(\varphi_: U_i \to X\) covering \(D\). For each index \(i\), we choose \(U_i\) small enough that \(\pi\) is one-to-one on \(U_i\), and we use \(\pi\) to identify \(U_i\) with a subset of \(M\). We have

where the equivalence relation \(\sim\) is given by

\[x_i \sim x_j \Longleftrightarrow g_{ij} \circ \varphi_i(x_i) = \varphi_j(x_j).\]*Here is the idea of the proof:* If \(\rho'\) is close to a holonomy representation \(\rho : \pi_1 M \to G\), then there are maps \(g'_{ij}\) close to the maps \(g_{ij}\) satisfying the cocycle condition. Then define

where the relation \(\sim'\) is defined by replacing \(g_{ij}\) with \(g'_{ij}\) above.

*Why can we do this?* Take the *nerve* of the cover, i.e., a graph \(H\) with one vertex for each set \(U_i\) and one edge between corresponding vertices whenever \(U_i \cap U_j\) is nonempty. We may assume without loss of generality that the sets \(U_i\) are round balls, so that \(H\) is connected. Now let \(T\) be a maximal tree with a chosen basepoint. Define \(g'_{ij}\) to be \(g_{ij}\) if the edge for \(g_{ij}\) is in \(T\); otherwise, choose \(g'_{ij}\) so that the based loop \(\alpha_{ij}\) in \(H\) containing the edge for \(g_{ij}\) is \(\rho(\alpha_{ij})\), a product of maps \(g_{kl}\) around the loop. \(\square\)

**Theorem** (Kozul 1962). *\(SC(M)\) is open in \(\mathrm{Hom}(\pi_1 M, \mathrm{PGL}(n + 1))\).*

*Proof.* Suppose we are given a strictly convex projective structure, and hence a holonomy representation \(\rho : \pi_1 M \to \mathrm{PGL}(n + 1)\) by the Proposition, and a nearby representation \(\rho'\). By the holonomy theorem, there exists a nearby projective structure, and this structure is a strictly convex structure (why?). \(\square\)

*HESSIAN METRICS*

Let \(U \subseteq \mathbb R^n\) be open, and let \(c : U \to \mathbb R^n\) be a smooth function that is strictly convex (i.e., whose Hessian matrix is positive definite). The map \(c\) induces a Riemannian metric on \(U\) given by

\[\langle u, v \rangle_c = u^\top A v \quad\text{for all }v, w \in T_x U \text{ and all } x \in U,\]where

\[A = D^2 c = {\left(\displaystyle{\frac{\partial^2 c}{\partial x_i \partial x_j}}\right)}\]is the Hessian matrix of \(c\).

For example, the map \(c : \mathbb R^2 \to \mathbb R\) given by \(c(x_1, x_2) = x_1^2 + x_2^2\) induces the Euclidean metric on \(\mathbb R^2\).

Recall that a nonzero tangent vector \(v \in T_x U\) is specified by a smooth curve \(\gamma : (-\epsilon, \epsilon) \to U\) with \(\gamma(0) = x\) and \(\gamma'(0) = v\), where \(x \in U\) and \(\epsilon > 0\). Then \(\|v\|_c^2 = F''(0) > 0\) by strict convexity, where \(F = c \circ \gamma\).

Let \(M\) be an affine \(n\)-manifold. Then we have charts \(\varphi_i : U_i \subseteq M \to \mathbb R^n\) and transition maps in the affine group \(\mathrm{Aff}(n)\).

Let \(c : M \to \mathbb R\) be smooth. Then

\[\begin{align*} c \text{ is strictly convex} &\Longleftrightarrow {} \text{all maps } c \circ \varphi_i^{-1} \text{ are strictly convex} \\ &\Longleftrightarrow {} (c \circ \gamma)''(t) > 0 \text{ for all } t \in (-\epsilon, \epsilon) \text{ for all curves } \gamma \text{ defined as above.} \end{align*}\]Thus \(c\) determines a Hessian metric on \(M\).

*Remark.* The Vinberg character functions on cones (April 14) are examples of this phenomenon.

**Theorem.** *Suppose \(M\) is a simply connected boundaryless affine \(n\)-manifold with a complete Hessian metric \(d\). Then the developing map \(M \to \mathbb R^n\) is injective and has image a convex set.*

*Proof.* Since \(M\) is simply connected, the developing map is a diffeomorphism, so we identify \(M\) with its image in \(\mathbb R^n\).

Let \(a, b \in M\) be distinct; we must show that the line segment joining them is contained in \(M\). Consider the triangle formed by \(a\), \(b\), and an arbitrary point \(p \in M\). Let \(\alpha, \beta : [0, 1] \to M\) be the straight-line paths paths from \(p\) to \(a\) and from \(p\) to \(b\), respectively. For each \(t \in [0, 1]\), let \(\gamma_t : [0, 1] \to M\) be the straight-line path from \(\alpha(t)\) to \(\beta(t)\). We claim that

\[\mathrm{length}(\gamma_t) \leq K < \infty\quad \text{for some } K > 0 \text{ for all } t.\]Then \(d(x, p) < L\) for some \(L > 0\) for all \(x \in M\), and we are done by completeness.

Let \(F_t = c \circ \gamma_t\) for each \(t\). We have

\[\mathrm{length}(\gamma_t) = \int^1_0 \sqrt{F''_t(s)}\,ds \leq {\left(\int^1_0 F''_t(s)\,ds\right)}^{1/2} {\left(\int^1_0 ds\right)}^{1/2} \leq (F'_t(1) + F'_t(0))^{1/2} \leq K.\]The first inequality follows from the Cauchy–Schwarz ineqality, and the last inequality (for some \(K > 0\)) follows from compactness. This proves the claim. \(\square\)

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