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

$\rho = h_* : \pi_1 M \stackrel{\cong}{\longrightarrow} \Gamma.$

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

$g_{ij} = \varphi_j \circ \varphi_i^{-1} : \varphi_i^{-1}(U_i \cap U_j) \longrightarrow \varphi_j^{-1}(U_i \cap U_j)$

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:

$g_{ik} = g_{jk} \circ g_{ij} \quad\text{whenever}\quad U_i \cap U_j \cap U_k \neq \emptyset.$

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

$M = \bigsqcup U_i \Big / \sim,$

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

$M' = \bigsqcup U_i \Big / \sim',$

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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